Are you struggling to solve the trigonometric ratios and functions in homework and assignments? Then, check out this **Big Ideas Math Algebra 2 Answers Chapter 9 Trigonometric Ratios and Functions** and solve your issues within seconds. The perfect way to learn the concepts of ch 9 is by practicing the questions given in the BIM Math Book Algebra 2 Chapter 9 Trigonometric Ratios and Functions Solution Key. Look no further and start your preparation by using this quick & ultimate guide. Constant practice can make you succeed in your math learning journey.

## Big Ideas Math Book Algebra 2 Answer Key Chapter 9 Trigonometric Ratios and Functions

Enhance your math skills by taking the help of our provided Big Ideas Math Book Algebra 2 Ch 9 Trigonometric Ratios and Functions Solution key. After learning the concepts of the Algebra 2 9th chapters from the **BIM Textbook Answers of Algebra 2,** students can become math proficient. Access the Topicwise Big Ideas Math Book Algebra 2 Ch 9 Trigonometric Ratios and Functions Answers for free from the respective links presented below and prepare well for the exams.

- Trigonometric Ratios and Functions Maintaining Mathematical Proficiency – Page 459
- Trigonometric Ratios and Functions Mathematical Practices – Page 460
- Lesson 9.1 Right Triangle Trigonometry – Page(461-468)
- Right Triangle Trigonometry 9.1 Exercises – Page(466-468)
- Lesson 9.2 Angles and Radian Measure – Page(469-476)
- Angles and Radian Measure 9.2 Exercises – Page(474-476)
- Lesson 9.3 Trigonometric Functions of Any Angle – Page(477-484)
- Trigonometric Functions of Any Angle 9.3 Exercises – Page(482-484)
- Lesson 9.4 Graphing Sine and Cosine Functions – Page(485-494)
- Graphing Sine and Cosine Functions 9.4 Exercises – Page(491-494)
- Trigonometric Ratios and Functions Study Skills: Form a Final Exam Study Group – Page 495
- Trigonometric Ratios and Functions 9.1–9.4 Quiz – Page 496
- Lesson 9.5 Graphing Other Trigonometric Functions – Page(497-504)
- Graphing Other Trigonometric Functions 9.5 Exercises – Page(502-504)
- Lesson 9.6 Modeling with Trigonometric Functions – Page(505-512)
- Modeling with Trigonometric Functions 9.6 Exercises – Page(510-512)
- Lesson 9.7 Using Trigonometric Identities – Page(513-518)
- Using Trigonometric Identities 9.7 Exercises – Page(517-518)
- Lesson 9.8 Using Sum and Difference Formulas – Page(519-524)
- Using Sum and Difference Formulas 9.8 Exercises – Page(523-524)
- Trigonometric Ratios and Functions Performance Task: Lightening the Load – Page 525
- Trigonometric Ratios and Functions Chapter Review – Page(526-530)
- Trigonometric Ratios and Functions Chapter Test – Page 531
- Trigonometric Ratios and Functions Cumulative Assessment – Page(532 – 533)

### Trigonometric Ratios and Functions Maintaining Mathematical Proficiency

**Order the expressions by value from least to greatest.**

Question 1.

log_{2 }x

∣4∣, 2 − 9∣, ∣6 + 4∣, − ∣7∣

Answer:

Question 2.

∣9 − 3∣, ∣0∣, ∣−4∣, \(\frac{|-5|}{|2|}\)

Answer:

Question 3.

∣−8^{3}∣,∣−2 • 8 ∣, ∣9 − 1∣, ∣9∣ + ∣−2∣ − ∣1 ∣

Answer:

Question 4.

∣−4 + 20∣, −∣4^{2}∣, ∣5∣−∣3 • 2 ∣, ∣−15∣

Answer:

**Find the missing side length of the triangle.**

Question 5.

Answer:

Question 6.

Answer:

Question 7.

Answer:

Question 8.

Answer:

Question 9.

Answer:

Question 10.

Answer:

Question 11.

**ABSTRACT REASONING**

The line segments connecting the points (x_{1}, y_{1}), (x_{2}, y_{1}), and (x_{2}, y_{2}) form a triangle. Is the triangle a right triangle? Justify your answer.

Answer:

### Trigonometric Ratios and Functions Mathematical Practices

Mathematically proficient students reason quantitatively by creating valid representations of problems.

**Monitoring Progress**

**Find the exact coordinates of the point (x, y) on the unit circle.**

Question 1.

Answer:

Question 2.

Answer:

Question 3.

Answer:

### Lesson 9.1 Right Triangle Trigonometry

**Essential Question** How can you find a trigonometric function of an acute angle θ?

Consider one of the acute angles θ of a right triangle. Ratios of a right triangle’s side lengths are used to define the six trigonometric functions, as shown.

**EXPLORATION 1**

Trigonometric Functions of Special Angles

Work with a partner. Find the exact values of the sine, cosine, and tangent functions for the angles 30°, 45°, and 60° in the right triangles shown.

**EXPLORATION 2**

Exploring Trigonometric Identities

Work with a partner.

Use the definitions of the trigonometric functions to explain why each trigonometric identity is true.

a. sin θ = cos(90° − θ)

b. cos θ = sin(90° − θ)

c. sin θ =\(\frac{1}{\csc \theta}\)

d. tan θ = \(\frac{1}{\cot \theta}\)

Use the definitions of the trigonometric functions to complete each trigonometric identity.

**Communicate Your Answer**

Question 3.

How can you find a trigonometric function of an acute angle θ?

Answer:

Question 4.

Use a calculator to find the lengths x and y of the legs of the right triangle shown.

Answer:

**Monitoring Progress**

**Evaluate the six trigonometric functions of the angle θ.**

Question 1.

Answer:

Question 2.

Answer:

Question 3.

Answer:

Question 4.

In a right triangle, θ is an acute angle and cos θ = \(\frac{7}{10}\). Evaluate the other five trigonometric functions of θ.

Answer:

Question 5.

Find the value of x for the right triangle shown.

Answer:

**Solve △ABC using the diagram at the left and the given measurements.**

Question 6.

B = 45°, c = 5

Answer:

Question 7.

A = 32°, b = 10

Answer:

Question 8.

A = 71°, c = 20

Answer:

Question 9.

B = 60°, a = 7

Answer:

Question 10.

In Example 5, find the distance between B and C.

Answer:

Question 11.

**WHAT IF?**

In Example 6, estimate the height of the parasailer above the boat when the angle of elevation is 38°.

Answer:

### Right Triangle Trigonometry 9.1 Exercises

**Vocabulary and Core Concept Check**

Question 1.

**COMPLETE THE SENTENCE**

In a right triangle, the two trigonometric functions of θ that are defined using the lengths of the hypotenuse and the side adjacent to θ are __________ and __________.

Answer:

Question 2.

**VOCABULARY**

Compare an angle of elevation to an angle of depression.

Answer:

Question 3.

**WRITING**

Explain what it means to solve a right triangle.

Answer:

Question 4.

**DIFFERENT WORDS, SAME QUESTION**

Which is different? Find “both” answers.

Answer:

**Monitoring Progress and Modeling with Mathematics**

**In Exercises 5–10, evaluate the six trigonometric functions of the angle θ.**

Question 5.

Answer:

Question 6.

Answer:

Question 7.

Answer:

Question 8.

Answer:

Question 9.

Answer:

Question 10.

Answer:

Question 11.

**REASONING**

Let θ be an acute angle of a right triangle. Use the two trigonometric functions tan θ = \(\frac{4}{9}\) and sec θ = \(\frac{\sqrt{97}}{9}\) to sketch and label the right triangle. Then evaluate the other four trigonometric functions of θ.

Answer:

Question 12.

**ANALYZING RELATIONSHIPS**

Evaluate the six trigonometric functions of the 90° − θ angle in Exercises 5–10. Describe the relationships you notice.

Answer:

**In Exercises 13–18, let θ be an acute angle of a right triangle. Evaluate the other five trigonometric functions of θ.**

Question 13.

sin θ = \(\frac{7}{11}\)

Answer:

Question 14.

cos θ = \(\frac{5}{12}\)

Answer:

Question 15.

tan θ = \(\frac{7}{6}\)

Answer:

Question 16.

csc θ = \(\frac{15}{8}\)

Answer:

Question 17.

sec θ = \(\frac{14}{9}\)

Answer:

Question 18.

cot θ = \(\frac{16}{11}\)

Answer:

Question 19.

**ERROR ANALYSIS**

Describe and correct the error in finding sin θ of the triangle below.

Answer:

Question 20.

**ERROR ANALYSIS**

Describe and correct the error in finding csc θ, given that θ is an acute angle of a right triangle and cos θ = \(\frac{7}{11}\).

Answer:

**In Exercises 21–26, find the value of x for the right triangle.**

Question 21.

Answer:

Question 22.

Answer:

Question 23.

Answer:

Question 24.

Answer:

Question 25.

Answer:

Question 26.

Answer:

**USING TOOLS **In Exercises 27–32, evaluate the trigonometric function using a calculator. Round your answer to four decimal places.

Question 27.

cos 14°

Answer:

Question 28.

tan 31°

Answer:

Question 29.

csc 59°

Answer:

Question 30.

sin 23°

Answer:

Question 31.

cot 6°

Answer:

Question 32.

sec 11°

Answer:

**In Exercises 33–40, solve △ABC using the diagram and the given measurements.**

Question 33.

B = 36°, a = 23

Answer:

Question 34.

A = 27°, b = 9

Answer:

Question 35.

A = 55°, a = 17

Answer:

Question 36.

B = 16°, b = 14

Answer:

Question 37.

A = 43°, b = 31

Answer:

Question 38.

B = 31°, a = 23

Answer:

Question 39.

B = 72°, c = 12.8

Answer:

Question 40.

A = 64°, a = 7.4

Answer:

Question 41.

**MODELING WITH MATHEMATICS**

To measure the width of a river, you plant a stake on one side of the river, directly across from a boulder. You then walk 100 meters to the right of the stake and measure a 79° angle between the stake and the boulder. What is the width w of the river?

Answer:

Question 42.

**MODELING WITH MATHEMATICS**

Katoomba Scenic Railway in Australia is the steepest railway in the world. The railway makes an angle of about 52° with the ground. The railway extends horizontally about 458 feet. What is the height of the railway?

Answer:

Question 43.

**MODELING WITH MATHEMATICS**

A person whose eye level is 1.5 meters above the ground is standing 75 meters from the base of the Jin Mao Building in Shanghai, China. The person estimates the angle of elevation to the top of the building is about 80°. What is the approximate height of the building?

Answer:

Question 44.

**MODELING WITH MATHEMATICS**

The Duquesne Incline in Pittsburgh, Pennsylvania, has an angle of elevation of 30°. The track has a length of about 800 feet. Find the height of the incline.

Answer:

Question 45.

**MODELING WITH MATHEMATICS**

You are standing on the Grand View Terrace viewing platform at Mount Rushmore, 1000 feet from the base of the monument.

a. You look up at the top of Mount Rushmore at an angle of 24°. How high is the top of the monument from where you are standing? Assume your eye level is 5.5 feet above the platform.

b. The elevation of the Grand View Terrace is 5280 feet. Use your answer in part (a) to find the elevation of the top of Mount Rushmore.

Answer:

Question 46.

**WRITING**

Write a real-life problem that can be solved using a right triangle. Then solve your problem.

Answer:

Question 47.

**MATHEMATICAL CONNECTIONS**

The Tropic of Cancer is the circle of latitude farthest north of the equator where the Sun can appear directly overhead. It lies 23.5° north of the equator, as shown.

a. Find the circumference of the Tropic of Cancer using 3960 miles as the approximate radius of Earth.

b. What is the distance between two points on the Tropic of Cancer that lie directly across from each other?

Answer:

Question 48.

**HOW DO YOU SEE IT?**

Use the figure to answer each question.

a. Which side is adjacent to θ?

b. Which side is opposite of θ?

c. Does cos θ = sin(90° − θ)? Explain.

Answer:

Question 49.

**PROBLEM SOLVING**

A passenger in an airplane sees two towns directly to the left of the plane.

a. What is the distance d from the airplane to the first town?

b. What is the horizontal distance x from the airplane to the first town?

c. What is the distance y between the two towns? Explain the process you used to find your answer.

Answer:

Question 50.

**PROBLEM SOLVING**

You measure the angle of elevation from the ground to the top of a building as 32°. When you move 50 meters closer to the building, the angle of elevation is 53°. What is the height of the building?

Answer:

Question 51.

**MAKING AN ARGUMENT**

Your friend claims it is possible to draw a right triangle so the values of the cosine function of the acute angles are equal. Is your friend correct? Explain your reasoning.

Answer:

Question 52.

**THOUGHT PROVOKING**

Consider a semicircle with a radius of 1 unit, as shown below. Write the values of the six trigonometric functions of the angle θ. Explain your reasoning.

Answer:

Question 53.

**CRITICAL THINKING**

A procedure for approximating π based on the work of Archimedes is to inscribe a regular hexagon in a circle.

a. Use the diagram to solve for x. What is the perimeter of the hexagon?

b. Show that a regular n-sided polygon inscribed in acircle of radius 1 has a perimeter of 2n • sin (\(\frac{180}{n}\))°.

c. Use the result from part (b) to find an expression in terms of n that approximates π. Then evaluate the expression when n= 50.

Answer:

**Maintaining Mathematical Proficiency.**

**Perform the indicated conversion.**

Question 54.

5 years to seconds

Answer:

Question 55.

12 pints to gallons

Answer:

Question 56.

5.6 meters to millimeters

Answer:

**Find the circumference and area of the circle with the given radius or diameter.**

Question 57.

r = 6 centimeters

Answer:

Question 58.

r = 11 inches

Answer:

Question 59.

d = 14 feet

Answer:

### Lesson 9.2 Angles and Radian Measure

**Essential Question** How can you find the measure of an angle in radians?

Let the vertex of an angle be at the origin, with one side of the angle on the positive x-axis. The radian measure of the angle is a measure of the intercepted arc length on a circle of radius 1. To convert between degree and radian measure, use the fact that \(\frac{\pi \text { radians }}{180^{\circ}}\) = 1.

**EXPLORATION 1**

Writing Radian Measures of Angles

Work with a partner. Write the radian measure of each angle with the given degree measure. Explain your reasoning.

**EXPLORATION 2**

Writing Degree Measures of Angles

Work with a partner. Write the degree measure of each angle with the given radian measure. Explain your reasoning.

**Communicate Your Answer**

Question 3.

How can you find the measure of an angle in radians?

Answer:

Question 4.

The figure shows an angle whose measure is 30 radians. What is the measure of the angle in degrees? How many times greater is 30 radians than 30 degrees? Justify your answers.

Answer:

**Monitoring Progress**

**Draw an angle with the given measure in standard position.**

Question 1.

65°

Answer:

Question 2.

300°

Answer:

Question 3.

−120°

Answer:

Question 4.

−450°

Answer:

**Find one positive angle and one negative angle that are coterminal with the given angle.**

Question 5.

80°

Answer:

Question 6.

230°

Answer:

Question 7.

740°

Answer:

Question 8.

−135°

Answer:

**Convert the degree measure to radians or the radian measure to degrees.**

Question 9.

135°

Answer:

Question 10.

−40°

Answer:

Question 11.

\(\frac{5 \pi}{4}\)

Answer:

Question 12.

−6.28

Answer:

Question 13.

**WHAT IF?**

In Example 4, the outfield fence is 220 feet from home plate. Estimate the length of the outfield fence and the area of the field.

Answer:

### Angles and Radian Measure 9.2 Exercises

**Vocabulary and Core Concept Check**

Question 1.

**COMPLETE THE SENTENCE**

An angle is in standard position when its vertex is at the __________ and its __________ lies on the positive x-axis.

Answer:

Question 2.

**WRITING**

Explain how the sign of an angle measure determines its direction of rotation.

Answer:

Question 3.

**VOCABULARY**

In your own words, define a radian.

Answer:

Question 4.

**WHICH ONE DOESN’T BELONG?**

Which angle does not belong with the other three? Explain your reasoning.

Answer:

**Monitoring Progress and Modeling with Mathematics**

**In Exercises 5–8, draw an angle with the given measure in standard position.**

Question 5.

110°

Answer:

Question 6.

450°

Answer:

Question 7.

−900°

Answer:

Question 8.

−10°

Answer:

**In Exercises 9–12, find one positive angle and one negative angle that are coterminal with the given angle.**

Question 9.

70°

Answer:

Question 10.

255°

Answer:

Question 11.

−125°

Answer:

Question 12.

−800°

Answer:

**In Exercises 13–20, convert the degree measure to radians or the radian measure to degrees.**

Question 13.

40°

Answer:

Question 14.

315°

Answer:

Question 15.

−260°

Answer:

Question 16.

−500°

Answer:

Question 17.

\(\frac{\pi}{9}\)

Answer:

Question 18.

\(\frac{3 \pi}{4}\)

Answer:

Question 19.

−5

Answer:

Question 20.

12

Answer:

Question 21.

**WRITING**

The terminal side of an angle in standard position rotates one-sixth of a revolution counterclockwise from the positive x-axis. Describe how to find the measure of the angle in both degree and radian measures.

Answer:

Question 22.

**OPEN-ENDED**

Using radian measure, give one positive angle and one negative angle that are coterminal with the angle shown. Justify your answers.

Answer:

**ANALYZING RELATIONSHIPS** In Exercises 23–26, match the angle measure with the angle.

Question 23.

600°

Answer:

Question 24.

\(-\frac{9 \pi}{4}\)

Answer:

Question 25.

\(\frac{5 \pi}{6}\)

Answer:

Question 26.

−240°

Answer:

Question 27.

**MODELING WITH MATHEMATICS**

The observation deck of a building forms a sector with the dimensions shown. Find the length of the safety rail and the area of the deck.

Answer:

Question 28.

**MODELING WITH MATHEMATICS**

In the men’s shot put event at the 2012 Summer Olympic Games, the length of the winning shot was 21.89 meters. A shot put must land within a sector having a central angle of 34.92° to be considered fair.

a. The officials draw an arc across the fair landing area, marking the farthest throw. Find the length of the arc.

b. All fair throws in the 2012 Olympics landed within a sector bounded by the arc in part (a). What is the area of this sector?

Answer:

Question 29.

**ERROR ANALYSIS**

Describe and correct the error in converting the degree measure to radians.

Answer:

Question 30.

**ERROR ANALYSIS**

Describe and correct the error in finding the area of a sector with a radius of 6 centimeters and a central angle of 40°.

Answer:

Question 31.

**PROBLEM SOLVING**

When a CD player reads information from the outer edge of a CD, the CD spins about 200 revolutions per minute. At that speed, through what angle does a point on the CD spin in oneminute? Give your answer in both degree and radian measures.

Answer:

Question 32.

**PROBLEM SOLVING**

You work every Saturday from 9:00 A.M. to 5:00 P.M. Draw a diagram that shows the rotation completed by the hour hand of a clock during this time. Find the measure of the angle generated by the hour hand in both degrees and radians. Compare this angle with the angle generated by the minute hand from 9:00 A.M. to 5:00 P.M.

Answer:

**USING TOOLS** In Exercises 33–38, use a calculator to evaluate the trigonometric function.

Question 33.

cos \(\frac{4 \pi}{3}\)

Answer:

Question 34.

sin \(\frac{7 \pi}{8}\)

Answer:

Question 35.

csc \(\frac{10 \pi}{11}\)

Answer:

Question 36.

cot (− \(\frac{6 \pi}{5}\))

Answer:

Question 37.

cot(−14)

Answer:

Question 38.

cos 6

Answer:

Question 39.

**MODELING WITH MATHEMATICS**

The rear windshield wiper of a car rotates 120°, as shown. Find the area cleared by the wiper.

Answer:

Question 40.

**MODELING WITH MATHEMATICS**

A scientist performed an experiment to study the effects of gravitational force on humans. In order for humans to experience twice Earth’s gravity, they were placed in a centrifuge 58 feet long and spun at a rate of about 15 revolutions per minute.

a. Through how many radians did the people rotate each second?

b. Find the length of the arc through which the people rotated each second.

Answer:

Question 41.

**REASONING**

In astronomy, the terminator is the day-night line on a planet that divides the planet into daytime and nighttime regions. The terminator moves across the surface of a planet as the planet rotates. It takes about 4 hours for Earth’s terminator to move across the continental United States. Through what angle has Earth rotated during this time? Give your answer in both degree and radian measures.

Answer:

Question 42.

**HOW DO YOU SEE IT?**

Use the graph to find the measure of θ. Explain your reasoning.

Answer:

Question 43.

**MODELING WITH MATHEMATICS**

A dartboard is divided into 20 sectors. Each sector is worth a point value from 1 to 20 and has shaded regions that double or triple this value. A sector is shown below. Find the areas of the entire sector, the double region, and the triple region.

Answer:

Question 44.

**THOUGHT PROVOKING**

π is an irrational number, which means that it cannot be written as the ratio of two whole numbers. π can, however, be written exactly as a continued fraction, as follows.

Answer:

Question 45.

**MAKING AN ARGUMENT**

Your friend claims that when the arc length of a sector equals the radius, the area can be given by A = \(\frac{s^{2}}{2}\). Is your friend correct? Explain.

Answer:

Question 46.

**PROBLEM SOLVING**

A spiral staircase has 15 steps. Each step is a sector with a radius of 42 inches and a central angle of \(\frac{\pi}{8}\).

a. What is the length of the arc formed by the outer edge of a step?

b. Through what angle would you rotate by climbing the stairs?

c. How many square inches of carpeting would you need to cover the 15 steps?

Answer:

Question 47.

**MULTIPLE REPRESENTATIONS**

There are 60 minutes in 1 degree of arc, and 60 seconds in 1 minute of arc. The notation 50° 30′ 10″ represents an angle with a measure of 50 degrees, 30 minutes, and 10 seconds.

a. Write the angle measure 70.55° using the notation above.

b. Write the angle measure 110° 45′ 30″ to the nearest hundredth of a degree. Justify your answer.

Answer:

**Maintaining Mathematical Proficiency**

**Find the distance between the two points.**

Question 48.

(1, 4), (3, 6)

Answer:

Question 49.

(−7, −13), (10, 8)

Answer:

Question 50.

(−3, 9), (−3, 16)

Answer:

Question 51.

(2, 12), (8, −5)

Answer:

Question 52.

(−14, −22), (−20, −32)

Answer:

Question 53.

(4, 16), (−1, 34)

Answer:

### Lesson 9.3 Trigonometric Functions of Any Angle

**Essential Question** How can you use the unit circle to define the trigonometric functions of any angle?

Let θ be an angle in standard position with (x, y) a point on the terminal side of θ and r = \(\sqrt{x^{2}+y^{2}}\) ≠ 0. The six trigonometric functions of θ are defined as shown.

**EXPLORATION 1**

Writing Trigonometric Functions

Work with a partner. Find the sine, cosine, and tangent of the angle θ in standard position whose terminal side intersects the unit circle at the point (x, y) shown.

**Communicate Your Answer**

Question 2.

How can you use the unit circle to define the trigonometric functions of any angle?

Answer:

Question 3.

For which angles are each function undefined? Explain your reasoning.

a. tangent

b. cotangent

c. secant

d. cosecant

Answer:

**Monitoring Progress**

**Evaluate the six trigonometric functions of θ.**

Question 1.

Answer:

Question 2.

Answer:

Question 3.

Answer:

Question 4.

Use the unit circle to evaluate the six trigonometric functions of θ = 180º.

Answer:

**Sketch the angle. Then find its reference angle.**

Question 5.

210°

Answer:

Question 6.

−260°

Answer:

Question 7.

\(\frac{-7 \pi}{9}\)

Answer:

Question 8.

\(\frac{15 \pi}{4}\)

Answer:

**Evaluate the function without using a calculator.**

Question 9.

cos(−210º)

Answer:

Question 10.

sec \(\frac{11 \pi}{4}\)

Answer:

Question 11.

Use the model given in Example 5 to estimate the horizontal distance traveled by a track and field long jumper who jumps at an angle of 20° and with an initial speed of 27 feet per second.

Answer:

### Trigonometric Functions of Any Angle 9.3 Exercises

**Vocabulary and Core Concept Check**

Question 1.

**COMPLETE THE SENTENCE**

A(n) ___________ is an angle in standard position whose terminal side lies on an axis.

Answer:

Question 2.

**WRITING**

Given an angle θ in standard position with its terminal side in Quadrant III, explain how you can use a reference angle to find cos θ.

Answer:

**Monitoring Progress and Modeling with Mathematics**

**In Exercises 3–8, evaluate the six trigonometric functions of θ.**

Question 3.

Answer:

Question 4.

Answer:

Question 5.

Answer:

Question 6.

Answer:

Question 7.

Answer:

Question 8.

Answer:

**In Exercises 9–14, use the unit circle to evaluate the six trigonometric functions of θ.**

Question 9.

θ = 0°

Answer:

Question 10.

θ = 540°

Answer:

Question 11.

θ = \(\frac{\pi}{2}\)

Answer:

Question 12.

θ = \(\frac{7 \pi}{2}\)

Answer:

Question 13.

θ = −270°

Answer:

Question 14.

θ = −2π

Answer:

**In Exercises 15–22, sketch the angle. Then find its reference angle.**

Question 15.

−100°

Answer:

Question 16.

150°

Answer:

Question 17.

320°

Answer:

Question 18.

−370°

Answer:

Question 19.

\(\frac{15 \pi}{4}\)

Answer:

Question 20.

\(\frac{8 \pi}{3}\)

Answer:

Question 21.

−\(\frac{5 \pi}{6}\)

Answer:

Question 22.

−\(\frac{13 \pi}{6}\)

Answer:

Question 23.

**ERROR ANALYSIS**

Let (−3, 2) be a point on the terminal side of an angle θ in standard position. Describe and correct the error in finding tan θ.

Answer:

Question 24.

**ERROR ANALYSIS**

Describe and correct the error in finding a reference angle θ′ for θ = 650°.

Answer:

**In Exercises 25–32, evaluate the function without using a calculator.**

Question 25.

sec 135°

Answer:

Question 26.

tan 240°

Answer:

Question 27.

sin(−150°)

Answer:

Question 28.

csc(−420°)

Answer:

Question 29.

tan (−\(\frac{3 \pi}{4}\))

Answer:

Question 30.

cot (\(\frac{-8 \pi}{3}\))

Answer:

Question 31.

cos \(\frac{7 \pi}{4}\)

Answer:

Question 32.

sec \(\frac{11 \pi}{6}\)

Answer:

**In Exercises 33–36, use the model for horizontal distance given in Example 5.**

Question 33.

You kick a football at an angle of 60° with an initial speed of 49 feet per second. Estimate the horizontal distance traveled by the football.

Answer:

Question 34.

The “frogbot” is a robot designed for exploring rough terrain on other planets. It can jump at a 45° angle with an initial speed of 14 feet per second. Estimate the horizontal distance the frogbot can jump on Earth.

Answer:

Question 35.

At what speed must the in-line skater launch himself off the ramp in order to land on the other side of the ramp?

Answer:

Question 36.

To win a javelin throwing competition, your last throw must travel a horizontal distance of at least 100 feet. You release the javelin at a 40° angle with an initial speed of 71 feet per second. Do you win the competition? Justify your answer.

Answer:

Question 37.

**MODELING WITH MATHEMATICS**

A rock climber is using a rock climbing treadmill that is 10 feet long. The climber begins by lying horizontally on the treadmill, which is then rotated about its midpoint by 110° so that the rock climber is climbing toward the top. If the midpoint of the treadmill is 6 feet above the ground, how high above the ground is the top of the treadmill?

Answer:

Question 38.

**REASONING**

A Ferris wheel has a radius of 75 feet. You board a car at the bottom of the Ferris wheel, which is 10 feet above the ground, and rotate 255°counterclockwise before the ride temporarily stops. How high above the ground are you when the ride stops? If the radius of the Ferris wheel is doubled, is your height above the ground doubled? Explain your reasoning.

Answer:

Question 39.

**DRAWING CONCLUSIONS**

A sprinkler at ground level is used to water a garden. The water leaving the sprinkler has an initial speed of 25 feet per second.

a. Use the model for horizontal distance given in Example 5 to complete the table.

b. Which value of θ appears to maximize the horizontal distance traveled by the water? Use the model for horizontal distance and the unit circle to explain why your answer makes sense.

c. Compare the horizontal distance traveled by the water when θ = (45 − k)° with the distance when θ = (45 + k)°, for 0 < k < 45.

Answer:

Question 40.

**MODELING WITH MATHEMATICS**

Your school’s marching band is performing at halftime during a football game. In the last formation, the band members form a circle 100 feet wide in the center of the field. You start at a point on the circle 100 feet from the goal line, march 300° around the circle, and then walk toward the goal line to exit the field. How far from the goal line are you at the point where you leave the circle?

Answer:

Question 41.

**ANALYZING RELATIONSHIPS**

Use symmetry and the given information to label the coordinates of the other points corresponding to special angles on the unit circle.

Answer:

Question 42.

**THOUGHT PROVOKING**

Use the interactive unit circle tool at BigIdeasMath.com to describe all values of θ for each situation.

a. sin θ > 0, cos θ < 0, and tan θ > 0

b. sin θ > 0, cos θ < 0, and tan θ < 0

Answer:

Question 43.

**CRITICAL THINKING**

Write tan θ as the ratio of two other trigonometric functions. Use this ratio to explain why tan 90° is undefined but cot 90° = 0.

Answer:

Question 44.

**HOW DO YOU SEE IT?**

Determine whether each of the six trigonometric functions of θ is positive, negative, or zero. Explain your reasoning.

Answer:

Question 45.

**USING STRUCTURE**

A line with slope m passes through the origin. An angle θ in standard position has a terminal side that coincides with the line. Use a trigonometric function to relate the slope of the line to the angle.

Answer:

Question 46.

**MAKING AN ARGUMENT**

Your friend claims that the only solution to the trigonometric equation tan θ = \(\sqrt{3}\) is θ= 60°. Is your friend correct? Explain your reasoning.

Answer:

Question 47.

**PROBLEM SOLVING**

When two atoms in a molecule are bonded to a common atom, chemists are interested in both the bond angle and the lengths of the bonds. An ozone molecule is made up of two oxygen atoms bonded to a third oxygen atom, as shown.

a. In the diagram, coordinates are given in picometers (pm). (Note: 1 pm = 10^{−12} m) Find the coordinates (x, y) of the center of the oxygen atom in Quadrant II.

b. Find the distance d (in picometers) between the centers of the two unbonded oxygen atoms.

Answer:

Question 48.

**MATHEMATICAL CONNECTIONS**

The latitude of a point on Earth is the degree measure of the shortest arc from that point to the equator. For example, the latitude of point P in the diagram equals the degree measure of arc PE. At what latitude θ is the circumference of the circle of latitude at P half the distance around the equator?

Answer:

**Maintaining Mathematical Proficiency**

**Find all real zeros of the polynomial function.**

Question 49.

f (x) = x^{4} + 2x^{3} + x^{2} + 8x − 12

Answer:

Question 50.

f(x) = x^{5} + 4x^{4} − 14x^{3} − 14x^{2} − 15x− 18

Answer:

Graph the function.

Question 51.

f(x) = 2(x+ 3)^{2} (x − 1)

Answer:

Question 52.

f(x) = \(\frac{1}{2}\) (x − 4)(x + 5)(x + 9)

Answer:

Question 53.

f(x) = x^{2}(x + 1)^{3} (x − 2)

Answer:

### Lesson 9.4 Graphing Sine and Cosine Functions

**Essential Question** What are the characteristics of the graphs of the sine and cosine functions?

**EXPLORATION 1**

Graphing the Sine FunctionWork with a partner.

a. Complete the table for y= sin x, where x is an angle measure in radians.

b. Plot the points (x, y) from part (a). Draw a smooth curve through the points to sketch the graph of y = sin x.

c. Use the graph to identify the x-intercepts, the x-values where the local maximums and minimums occur, and the intervals for which the function is increasing or decreasing over −2π ≤ x ≤ 2π. Is the sine function even, odd, or neither?

**EXPLORATION 2**

Graphing the Cosine Function

Work with a partner.

a. Complete a table for y= cos x using the same values of x as those used in Exploration 1.

b. Plot the points (x, y) from part (a) and sketch the graph of y= cos x.

c. Use the graph to identify the x-intercepts, the x-values where the local maximums and minimums occur, and the intervals for which the function is increasing or decreasing over −2π ≤ x ≤ 2π. Is the cosine function even, odd, or neither?

**Communicate Your Answer**

Question 3.

What are the characteristics of the graphs of the sine and cosine functions?

Answer:

Question 4.

Describe the end behavior of the graph of y = sin x

Answer:

**Monitoring Progress**

Identify the amplitude and period of the function. Then graph the function and describe the graph of g as a transformation of the graph of its parent function.

Question 1.

g(x) = \(\frac{1}{4}\)sin x

Answer:

Question 2.

g(x) = cos 2x

Answer:

Question 3.

g(x) = 2 sin πx

Answer:

Question 4.

g(x) = \(\frac{1}{3}\) cos \(\frac{1}{2}\)x

Answer:

**Graph the function.**

Question 5.

g(x) = cos x+ 4

Answer:

Question 6.

g(x) = \(\frac{1}{2}\)sin (x − \(\left.\frac{\pi}{2}\right\))

Answer:

Question 7.

g(x) = sin(x + π) − 1

Answer:

**Graph the function.**

Question 8.

g(x) = −cos (x + \(\left.\frac{\pi}{2}\right\))

Answer:

Question 9.

g(x) = −3 sin \(\frac{1}{2}\)x + 2

Answer:

Question 10.

g(x) = −2 cos 4x − 1

Answer:

### Graphing Sine and Cosine Functions 9.4 Exercises

**Vocabulary and Core Concept Check**

Question 1.

**COMPLETE THE SENTENCE**

The shortest repeating portion of the graph of a periodic function is called a(n) _________.

Answer:

Question 2.

**WRITING**

Compare the amplitudes and periods of the functions y = \(\frac{1}{2}\)cos x and y = 3 cos 2x.

Answer:

Question 3.

**VOCABULARY**

What is a phase shift? Give an example of a sine function that has a phase shift.

Answer:

Question 4.

**VOCABULARY**

What is the midline of the graph of the function y = 2 sin 3(x + 1) − 2?

Answer:

**Monitoring Progress and Modeling with Mathematics**

**USING STRUCTURE** In Exercises 5–8, determine whether the graph represents a periodic function. If so, identify the period.

Question 5.

Answer:

Question 6.

Answer:

Question 7.

Answer:

Question 8.

Answer:

**In Exercises 9–12, identify the amplitude and period of the graph of the function.**

Question 9.

Answer:

Question 10.

Answer:

Question 11.

Answer:

Question 12.

Answer:

**In Exercises 13–20, identify the amplitude and period of the function. Then graph the function and describe the graph of g as a transformation of the graph of its parent function.**

Question 13.

g(x) = 3 sin x

Answer:

Question 14.

g(x) = 2 sin x

Answer:

Question 15.

g(x) = cos 3x

Answer:

Question 16.

g(x) = cos 4x

Answer:

Question 17.

g(x) = sin 2πx

Answer:

Question 18.

g(x) = 3 sin 2x

Answer:

Question 19.

g(x) = \(\frac{1}{3}\)cos 4x

Answer:

Question 20.

g(x) = \(\frac{1}{2}\)cos 4πx

Answer:

Question 21.

**ANALYZING EQUATIONS**

Which functions have an amplitude of 4 and a period of 2?

A. y = 4 cos 2x

B. y = −4 sin πx

C. y = 2 sin 4x

D. y = 4 cos πx

Answer:

Question 22.

**WRITING EQUATIONS**

Write an equation of the form y = a sin bx, where a > 0 and b > 0, so that the graph has the given amplitude and period.

a. amplitude: 1

period: 5

b. amplitude: 10

period: 4

c. amplitude: 2

period: 2π

d. amplitude: \(\frac{1}{2}\)

period: 3π

Answer:

Question 23.

**MODELING WITH MATHEMATICS**

The motion of a pendulum can be modeled by the function d = 4 cos 8πt, where d is the horizontal displacement (in inches) of the pendulum relative to its position at rest and t is the time (in seconds). Find and interpret the period and amplitude in the context of this situation. Then graph the function.

Answer:

Question 24.

**MODELING WITH MATHEMATICS**

A buoy bobs up and down as waves go past. The vertical displacement y (in feet) of the buoy with respect to sea level can be modeled by y = 1.75 cos \(\frac{\pi}{3}\)t, where t is the time (in seconds). Find and interpret the period and amplitude in the context of the problem. Then graph the function.

Answer:

**In Exercises 25–34, graph the function.**

Question 25.

g(x) = sin x + 2

Answer:

Question 26.

g(x) = cos x − 4

Answer:

Question 27.

g(x) = cos (x − \(\frac{\pi}{2}\))

Answer:

Question 28.

g(x) = sin (x + \(\frac{\pi}{4}\))

Answer:

Question 29.

g(x) = 2 cos x − 1

Answer:

Question 30.

g(x) = 3 sin x + 1

Answer:

Question 31.

g(x) = sin 2(x + π)

Answer:

Question 32.

g(x) = cos 2(x − π)

Answer:

Question 33.

g(x) = sin \(\frac{1}{2}\)(x + 2π) + 3

Answer:

Question 34.

g(x) = cos \(\frac{1}{2}\)(x − 3π) − 5

Answer:

Question 35.

**ERROR ANALYSIS**

Describe and correct the error in finding the period of the function y = sin \(\frac{2}{3}\)x.

Answer:

Question 36.

**ERROR ANALYSIS**

Describe and correct the error in determining the point where the maximum value of the function y = 2 sin (x − \(\frac{\pi}{4}\)) occurs.

Answer:

**USING STRUCTURE** In Exercises 37–40, describe the transformation of the graph of f represented by the function g

Question 37.

f(x) = cos x, g(x) = 2 cos (x − \(\frac{\pi}{2}\)) + 1

Answer:

Question 38.

f(x) = sin x, g(x) = 3 sin (x + \(\frac{\pi}{4}\)) − 2

Answer:

Question 39.

f(x) = sin x, g(x) = sin 3(x + 3π) − 5

Answer:

Question 40.

f(x) = cos x, g(x) = cos 6(x − π) + 9

Answer:

**In Exercises 41–48, graph the function.**

Question 41.

g(x) = −cos x + 3

Answer:

Question 42.

g(x) = −sin x − 5

Answer:

Question 43.

g(x) = −sin \(\frac{1}{2}\)x − 2

Answer:

Question 44.

g(x) = −cos 2x + 1

Answer:

Question 45.

g(x) = −sin(x − π) + 4

Answer:

Question 46.

g(x) = −cos(x + π) − 2

Answer:

Question 47.

g(x) = −4 cos (x + \(\frac{\pi}{4}\)) − 1

Answer:

Question 48.

g(x) = −5 sin (x − \(\frac{\pi}{2}\)) + 3

Answer:

Question 49.

**USING EQUATIONS**

Which of the following is a point where the maximum value of the graph of y =−4 cos (x − \(\frac{\pi}{2}\))occurs?

A. (−\(\frac{\pi}{2}\), 4 )

B. (\(\frac{\pi}{2}\), 4 )

C. (0, 4)

D. (π, 4)

Answer:

Question 50.

**ANALYZING RELATIONSHIPS**

Match each function with its graph. Explain your reasoning.

a. y = 3 + sin x

b. y = −3 + cos x

c. y = sin 2(x − \(\frac{\pi}{2}\))

d. y = cos 2 (x − \(\frac{\pi}{2}\))

Answer:

**WRITING EQUATIONS** In Exercises 51–54, write a rule for g that represents the indicated transformations of the graph of f.

Question 51.

f(x) = 3 sin x; translation 2 units up and π units right

Answer:

Question 52.

f(x) = cos 2πx; translation 4 units down and 3 units left

Answer:

Question 53.

f(x) = \(\frac{1}{3}\)cos πx; translation 1 unit down, followed by a reflection in the line y =−1

Answer:

Question 54.

f(x) = \(\frac{1}{2}\) sin 6x; translation \(\frac{3}{2}\) units down and 1 unit right, followed by a reflection in the line y = −\(\frac{3}{2}\)

Answer:

Question 55.

**MODELING WITH MATHEMATICS**

The height h(in feet) of a swing above the ground can be modeled by the function h = −8 cos θ+ 10, where the pivot is 10 feet above the ground, the rope is 8 feet long, and θ is the angle that the rope makes with the vertical. Graph the function. What is the height of the swing when θ is 45°?

Answer:

Question 56.

**DRAWING A CONCLUSION**

In a particular region, the population L (in thousands) of lynx (the predator) and the population H (in thousands) of hares (the prey) can be modeled by the equations

L = 11.5 + 6.5 sin\(\frac{\pi}{5}\)t

H = 27.5 + 17.5 cos \(\frac{\pi}{5}\)t

where t is the time in years.

a. Determine the ratio of hares to lynx when t = 0, 2.5, 5, and 7.5 years.

b. Use the figure to explain how the changes in the two populations appear to be related.

Answer:

Question 57.

**USING TOOLS**

The average wind speed s (in miles per hour) in the Boston Harbor can be approximated by s = 3.38 sin\(\frac{\pi}{180}\)(t + 3) + 11.6 where t is the time in days and t = 0 represents January 1. Use a graphing calculator to graph the function. On which days of the year is the average wind speed 10 miles per hour? Explain your reasoning.

Answer:

Question 58.

**USING TOOLS**

The water depth d (in feet) for the Bay of Fundy can be modeled by d = 35 − 28 cos \(\frac{\pi}{6.2}\)t, where t is the time in hours and t = 0 represents midnight. Use a graphing calculator to graph the function. At what time(s) is the water depth 7 feet? Explain.

Answer:

Question 59.

**MULTIPLE REPRESENTATIONS**

Find the average rate of change of each function over the interval 0 < x < π.

a. y = 2 cos x

Answer:

Question 60.

**REASONING**

Consider the functions y = sin(−x) and y = cos(−x).

a. Construct a table of values for each equation using the quadrantal angles in the interval −2π ≤ x ≤ 2π.

b. Graph each function.

c. Describe the transformations of the graphs of the parent functions.

Answer:

Question 61.

**MODELING WITH MATHEMATICS**

You are riding a Ferris wheel that turns for 180 seconds. Your height h (in feet) above the ground at any time t (in seconds) can be modeled by the equation h = 85 sin\(\frac{\pi}{20}\)(t − 10) + 90.

a. Graph the function.

b. How many cycles does the Ferris wheel make in 180 seconds?

c. What are your maximum and minimum heights?

Answer:

Question 62.

**HOW DO YOU SEE IT?**

Use the graph to answer each question.

a. Does the graph represent a function of the form f(x) = a sin bx or f(x) =a cos bx? Explain.

b. Identify the maximum value, minimum value, period, and amplitude of the function.

Answer:

Question 63.

**FINDING A PATTERN**

Write an expression in terms of the integer n that represents all the x-intercepts of the graph of the function y = cos 2x. Justify your answer.

Answer:

Question 64.

**MAKING AN ARGUMENT**

Your friend states that for functions of the form y = a sin bx and y = a cos bx, the values of a and b affect the x-intercepts of the graph of the function. Is your friend correct? Explain.

Answer:

Question 65.

**CRITICAL THINKING**

Describe a transformation of the graph of f(x) = sin x that results in the graph of g(x) = cos x.

Answer:

Question 66.

**THOUGHT PROVOKING**

Use a graphing calculator to find a function of the form y = sin b_{1}x + cos b_{2}xwhose graph matches that shown below.

Answer:

Question 67.

**PROBLEM SOLVING**

For a person at rest, the blood pressure P (in millimeters of mercury) at time t (in seconds) is given by the function

P = 100 − 20 cos \(\frac{8 \pi}{3}\)t.

Graph the function. One cycle is equivalent to one heartbeat. What is the pulse rate (in heartbeats per minute) of the person?

Answer:

Question 68.

**PROBLEM SOLVING**

The motion of a spring can be modeled by y = A cos kt, where y is the vertical displacement (in feet) of the spring relative to its position at rest, A is the initial displacement (in feet), k is a constant that measures the elasticity of the spring, and t is the time (in seconds).

a. You have a spring whose motion can be modeled by the function y= 0.2 cos 6t. Find the initial displacement and the period of the spring. Then graph the function.

b. When a damping force is applied to the spring, the motion of the spring can be modeled by the function y = 0.2e^{−4.5t} cos 4t. Graph this function. What effect does damping have on the motion?

Answer:

**Maintaining Mathematical Proficiency**

**Simplify the rational expression, if possible.**

Question 69.

\(\frac{x^{2}+x-6}{x+3}\)

Answer:

Question 70.

\(\frac{x^{3}-2 x^{2}-24 x}{x^{2}-2 x-24}\)

Answer:

Question 71.

\(\frac{x^{2}-4 x-5}{x^{2}+4 x-5}\)

Answer:

Question 72.

\(\frac{x^{2}-16}{x^{2}+x-20}\)

Answer:

**Find the least common multiple of the expressions.**

Question 73.

2x, 2(x − 5)

Answer:

Question 74.

x^{2} − 4, x + 2

Answer:

Question 75.

x^{2} + 8x + 12, x + 6

Answer:

### Trigonometric Ratios and Functions Study Skills: Form a Final Exam Study Group

**9.1–9.4 What Did You Learn?**

**Core Vocabulary**

**Core Concepts**

**Mathematical Practices**

Question 1.

Make a conjecture about the horizontal distances traveled in part (c) of Exercise 39 on page 483.

Answer:

Question 2.

Explain why the quantities in part (a) of Exercise 56 on page 493 make sense in the context of the situation.

Answer:

**Study Skills: Form a Final Exam Study Group**

Form a study group several weeks before the final exam. The intent of this group is to review what you have already learned while continuing to learn new material.

### Trigonometric Ratios and Functions 9.1–9.4 Quiz

Question 1.

In a right triangle, θ is an acute angle and sin θ = \(\frac{1}{2}\). Evaluate the other five trigonometric functions of θ.

Answer:

**Find the value of x for the right triangle.**

Question 2.

Answer:

Question 3.

Answer:

Question 4.

Answer:

Draw an angle with the given measure in standard position. Then find one positive angle and one negative angle that are coterminal with the given angle.

Question 5.

40°

Answer:

Question 6.

\(\frac{5 \pi}{6}\)

Answer:

Question 7.

−960°

Answer:

**Convert the degree measure to radians or the radian measure to degrees.**

Question 8.

\(\frac{3 \pi}{10}\)

Answer:

Question 9.

−60°

Answer:

Question 10.

72°

Answer:

**Evaluate the six trigonometric functions of θ.**

Question 11.

Answer:

Question 12.

Answer:

Question 13.

Answer:

Question 14.

Identify the amplitude and period of g(x) = 3 sin x. Then graph the function and describe the graph of g as a transformation of the graph of f (x) = sin x.

Answer:

Question 15.

Identify the amplitude and period of g(x) = cos 5πx + 3. Then graph the function and describe the graph of g as a transformation of the graph of f(x) = cos x.

Answer:

Question 16.

You are flying a kite at an angle of 70°. You have let out a total of 400 feet of string and are holding the reel steady 4 feet above the ground.

a. How high above the ground is the kite?

b. A friend watching the kite estimates that the angle of elevation to the kite is 85°. How far from your friend are you standing?

Answer:

Question 17.

The top of the Space Needle in Seattle, Washington, is a revolving, circular restaurant. The restaurant has a radius of 47.25 feet and makes one complete revolution in about an hour. You have dinner at a window table from 7:00 P.M. to 8:55 P.M. Compare the distance you revolve with the distance of a person seated 5 feet away from the windows.

Answer:

### Lesson 9.5 Graphing Other Trigonometric Functions

**Essential Question** What are the characteristics of the graph of the tangent function?

**EXPLORATION 1**

Graphing the Tangent Function

Work with a partner. a. Complete the table for y = tan x, where x is an angle measure in radians.

b. The graph of y = tan x has vertical asymptotes at x-values where tan x is undefined. Plot the points (x, y) from part (a). Then use the asymptotes to sketch the graph of y = tan x.

c. For the graph of y = tan x, identify the asymptotes, the x-intercepts, and the intervals for which the function is increasing or decreasing over −\(\frac{\pi}{2}\) ≤ x ≤ \(\frac{3 \pi}{2}\). Is the tangent function even, odd, or neither?

**Communicate Your Answer**

Question 2.

What are the characteristics of the graph of the tangent function?

Answer:

Question 3.

Describe the asymptotes of the graph of y = cot x on the interval −\(\frac{\pi}{2}\) < x < \(\frac{3 \pi}{2}\).

Answer:

**Monitoring Progress**

**Graph one period of the function. Describe the graph of g as a transformation of the graph of its parent function.**

Question 1.

g(x) = tan 2x

Answer:

Question 2.

g(x) = \(\frac{1}{3}\)cot x

Answer:

Question 3.

g(x) = 2 cot 4x

Answer:

Question 4.

g(x) = 5 tan πx

Answer:

**Graph one period of the function. Describe the graph of g as a transformation of the graph of its parent function.**

Question 5.

g(x) = csc 3x

Answer:

Question 6.

g(x) = \(\frac{1}{2}\)sec x

Answer:

Question 7.

g(x) = 2 csc 2x

Answer:

Question 8.

g(x) = 2 sec πx

Answer:

### Graphing Other Trigonometric Functions 9.5 Exercises

**Vocabulary and Core Concept Check**

Question 1.

**WRITING**

Explain why the graphs of the tangent, cotangent, secant, and cosecant functions do not have an amplitude.

Answer:

Question 2.

**COMPLETE THE SENTENCE**

The _______ and _______ functions are undefined for x-values at which sin x = 0.

Answer:

Question 3.

**COMPLETE THE SENTENCE**

The period of the function y = sec x is _____, and the period of y = cot x is _____.

Answer:

Question 4.

**WRITING**

Explain how to graph a function of the form y = a sec bx.

Answer:

**Monitoring Progress and Modeling with Mathematics**

**In Exercises 5–12, graph one period of the function. Describe the graph of gas a transformation of the graph of its parent function.**

Question 5.

g(x) = 2 tan x

Answer:

Question 6.

g(x) = 3 tan x

Answer:

Question 7.

g(x) = cot 3x

Answer:

Question 8.

g(x) = cot 2x

Answer:

Question 9.

g(x) = 3 cot \(\frac{1}{4}\)x

Answer:

Question 10.

g(x) = 4 cot\(\frac{1}{2}\)x

Answer:

Question 11.

g(x) = \(\frac{1}{2}\)tan πx

Answer:

Question 12.

g(x) = \(\frac{1}{3}\) tan 2πx

Answer:

Question 13.

**ERROR ANALYSIS**

Describe and correct the error in finding the period of the function y = cot 3x.

Answer:

Question 14.

**ERROR ANALYSIS**

Describe and correct the error in describing the transformation of f(x) = tan x represented by g(x) = 2 tan 5x.

Answer:

Question 15.

**ANALYZING RELATIONSHIPS**

Use the given graph to graph each function.

Answer:

Question 16.

**USING EQUATIONS**

Which of the following are asymptotes of the graph of y = 3 tan 4x?

A. x = \(\frac{\pi}{8}\)

B. x = \(\frac{\pi}{4}\)

C. x = 0

D. x = −\(\frac{5 \pi}{8}\)

Answer:

In Exercises 17–24, graph one period of the function. Describe the graph of gas a transformation of the graph of its parent function.

Question 17.

g(x) = 3 csc x

Answer:

Question 18.

g(x) = 2 csc x

Answer:

Question 19.

g(x) = sec 4x

Answer:

Question 20.

g(x) = sec 3x

Answer:

Question 21.

g(x) = \(\frac{1}{2}\)sec πx

Answer:

Question 22.

g(x) = \(\frac{1}{4}\) sec 2πx

Answer:

Question 23.

g(x) = csc \(\frac{\pi}{2}\)x

Answer:

Question 24.

g(x) = csc \(\frac{\pi}{4}\)x

Answer:

**ATTENDING TO PRECISION** In Exercises 25–28, use the graph to write a function of the form y = a tan bx.

Question 25.

Answer:

Question 26.

Answer:

Question 27.

Answer:

Question 28.

Answer:

**USING STRUCTURE** In Exercises 29–34, match the equation with the correct graph. Explain your reasoning.

Question 29.

g(x) = 4 tan x

Answer:

Question 30.

g(x) = 4 cot x

Answer:

Question 31.

g(x) = 4 csc πx

Answer:

Question 32.

g(x) = 4 sec πx

Answer:

Question 33.

g(x) = sec 2x

Answer:

Question 34.

g(x) = csc 2x

Answer:

Question 35.

**WRITING**

Explain why there is more than one tangent function whose graph passes through the origin and has asymptotes at x = −π and x = π.

Answer:

Question 36.

**USING EQUATIONS**

Graph one period of each function. Describe the transformation of the graph of its parent function.

a. g(x) = sec x + 3

b. g(x) = csc x − 2

c. g(x) = cot(x − π)

d. g(x) = −tan x

Answer:

**WRITING EQUATIONS** In Exercises 37–40, write a rule for g that represents the indicated transformation of the graph of f.

Question 37.

f(x) = cot 2x; translation 3 units up and \(\frac{\pi}{2}\) units left

Answer:

Question 38.

f(x) = 2 tan x; translation π units right, followed by a horizontal shrink by a factor of \(\frac{1}{3}\)

Answer:

Question 39.

f(x) = 5 sec (x − π); translation 2 units down, followed by a reflection in the x-axis

Answer:

Question 40.

f(x) = 4 csc x; vertical stretch by a factor of 2 and a reflection in the x-axis

Answer:

Question 41.

**MULTIPLE REPRESENTATIONS**

Which function has a greater local maximum value? Which has a greater local minimum value? Explain.

Answer:

Question 42.

**ANALYZING RELATIONSHIPS**

Order the functions from the least average rate of change to the greatest average rate of change over the interval −\(\frac{\pi}{4}\) < x < \(\frac{\pi}{4}\).

Answer:

Question 43.

**REASONING**

You are standing on a bridge 140 feet above the ground. You look down at a car traveling away from the underpass. The distance d (in feet) the car is from the base of the bridge can be modeled by d= 140 tan θ. Graph the function. Describe what happens to θ as d increases

Answer:

Question 44.

**USING TOOLS**

You use a video camera to pan up the Statue of Liberty. The height h (in feet) of the part of the Statue of Liberty that can be seen through your video camera after time t (in seconds) can be modeled by h= 100 tan \(\frac{\pi}{36}\)t. Graph the function using a graphing calculator. What viewing window did you use? Explain.

Answer:

Question 45.

**MODELING WITH MATHEMATICS**

You are standing 120 feet from the base of a 260-foot building. You watch your friend go down the side of the building in a glass elevator.

a. Write an equation that gives the distance d (in feet) your friend is from the top of the building as a function of the angle of elevation θ.

b. Graph the function found in part (a). Explain how the graph relates to this situation.

Answer:

Question 46.

**MODELING WITH MATHEMATICS**

You are standing 300 feet from the base of a 200-foot cliff. Your friend is rappelling down the cliff.

a. Write an equation that gives the distance d(in feet) your friend is from the top of the cliff as a function of the angle of elevation θ.

b. Graph the function found in part (a).

c. Use a graphing calculator to determine the angle of elevation when your friend has rappelled halfway down the cliff.

Answer:

Question 47.

**MAKING AN ARGUMENT**

Your friend states that it is not possible to write a cosecant function that has the same graph as y = sec x. Is your friend correct? Explain your reasoning.

Answer:

Question 48.

**HOW DO YOU SEE IT?**

Use the graph to answer each question.

a. What is the period of the graph?

b. What is the range of the function?

c. Is the function of the form f(x) = a csc bx or f(x) = a sec bx? Explain.

Answer:

Question 49.

**ABSTRACT REASONING**

Rewrite a sec bx in terms of cos bx. Use your results to explain the relationship between the local maximums and minimums of the cosine and secant functions.

Answer:

Question 50.

**THOUGHT PROVOKING**

A trigonometric equation that is true for all values of the variable for which both sides of the equation are defined is called a trigonometric identity.Use a graphing calculator to graph the function

y = \(\frac{1}{2}\)(tan \(\frac{x}{2}\) + cot \(\frac{x}{2}\)) .

Use your graph to write a trigonometric identity involving this function. Explain your reasoning.

Answer:

Question 51.

**CRITICAL THINKING**

Find a tangent function whose graph intersects the graph of y = 2 + 2 sin x only at minimum points of the sine function.

Answer:

**Maintaining Mathematical Proficiency**

**Write a cubic function whose graph passes through the given points.**

Question 52.

(−1, 0), (1, 0), (3, 0), (0, 3)

Answer:

Question 53.

(−2, 0), (1, 0), (3, 0), (0, −6)

Answer:

Question 54.

(−1, 0), (2, 0), (3, 0), (1, −2)

Answer:

Question 55.

(−3, 0), (−1, 0), (3, 0), (−2, 1)

Answer:

**Find the amplitude and period of the graph of the function.**

Question 56.

Answer:

Question 57.

Answer:

Question 58.

Answer:

### Lesson 9.6 Modeling with Trigonometric Functions

**Essential Question** What are the characteristics of the real-life problems that can be modeled by trigonometric functions?

**EXPLORATION 1**

Modeling Electric Currents

Work with a partner. Find a sine function that models the electric current shown in each oscilloscope screen. State the amplitude and period of the graph.

**Communicate Your Answer**

Question 2.

What are the characteristics of the real-life problems that can be modeled by trigonometric functions?

Answer:

Question 3.

Use the Internet or some other reference to find examples of real-life situations that can be modeled by trigonometric functions.

Answer:

**Monitoring Progress**

Question 1.

WHAT IF?

In Example 1, how would the function change when the audiometer produced a pure tone with a frequency of 1000 hertz?

Answer:

**Write a function for the sinusoid.**

Question 2.

Answer:

Question 3.

Answer:

Question 4.

**WHAT IF?**

Describe how the model in Example 3 changes when the lowest point of a rope is 5 inches above the ground and the highest point is 70 inches above the ground.

Answer:

Question 5.

The table shows the average daily temperature T (in degrees Fahrenheit) for a city each month, where m = 1 represents January. Write a model that gives T as a function of m and interpret the period of its graph.

Answer:

### Modeling with Trigonometric Functions 9.6 Exercises

**Vocabulary and Core Concept Check**

Question 1.

**COMPLETE THE SENTENCE**

Graphs of sine and cosine functions are called __________.

Answer:

Question 2.

**WRITING**

Describe how to find the frequency of the function whose graph is shown.

Answer:

**Monitoring Progress and Modeling with Mathematics**

**In Exercises 3–10, find the frequency of the function.**

Question 3.

y = sin x

Answer:

Question 4.

y = sin 3x

Answer:

Question 5.

y = cos 4x + 2

Answer:

Question 6.

y =−cos 2x

Answer:

Question 7.

y = sin 3πx

Answer:

Question 8.

y = cos \(\frac{\pi x}{4}\)

Answer:

Question 9.

y = \(\frac{1}{2}\) cos 0.75x − 8

Answer:

Question 10.

y = 3 sin 0.2x + 6

Answer:

Question 11.

**MODELING WITH MATHEMATICS**

The lowest frequency of sounds that can be heard by humans is 20 hertz. The maximum pressure P produced from a sound with a frequency of 20 hertz is 0.02 millipascal. Write and graph a sine model that gives the pressure P as a function of the time t(in seconds).

Answer:

Question 12.

**MODELING WITH MATHEMATICS**

A middle-A tuning fork vibrates with a frequency f of 440 hertz (cycles per second). You strike a middle-A tuning fork with a force that produces a maximum pressure of 5 pascals. Write and graph a sine model that gives the pressure Pas a function of the time t (in seconds).

Answer:

**In Exercises 13–16, write a function for the sinusoid.**

Question 13.

Answer:

Question 14.

Answer:

Question 15.

Answer:

Question 16.

Answer:

Question 17.

**ERROR ANALYSIS**

Describe and correct the error in finding the amplitude of a sinusoid with a maximum point at (2, 10) and a minimum point at (4, −6).

Answer:

Question 18.

**ERROR ANALYSIS**

Describe and correct the error in finding the vertical shift of a sinusoid with a maximum point at (3, −2) and a minimum point at (7, −8).

Answer:

Question 19.

**MODELING WITH MATHEMATICS**

One of the largest sewing machines in the world has a flywheel (which turns as the machine sews) that is 5 feet in diameter. The highest point of the handle at the edge of the flywheel is 9 feet above the ground, and the lowest point is 4 feet. The wheel makes a complete turn every 2 seconds. Write a model for the height h(in feet) of the handle as a function of the time t(in seconds) given that the handle is at its lowest point when t = 0.

Answer:

Question 20.

**MODELING WITH MATHEMATICS**

The Great LaxeyWheel, located on the Isle of Man, is the largest working water wheel in the world. The highest point of a bucket on the wheel is 70.5 feet above the viewing platform, and the lowest point is 2 feet below the viewing platform. The wheel makes a complete turn every 24 seconds. Write a model for the height h(in feet) of the bucket as a function of time t (in seconds) given that the bucket is at its lowest point when t = 0.

Answer:

**USING TOOLS** In Exercises 21 and 22, the time t is measured in months, where t = 1 represents January. Write a model that gives the average monthly high temperature D as a function of t and interpret the period of the graph.

Question 21.

Answer:

Question 22.

Answer:

Question 23.

**MODELING WITH MATHEMATICS**

A circuit has an alternating voltage of 100 volts that peaks every 0.5 second. Write a sinusoidal model for the voltage Vas a function of the time t (in seconds).

Answer:

Question 24.

**MULTIPLE REPRESENTATIONS**

The graph shows the average daily temperature of Lexington, Kentucky. The average daily temperature of Louisville, Kentucky, is modeled by y =−22 cos \(\frac{\pi}{6}\)t + 57, where y is the temperature (in degrees Fahrenheit) and t is the number of months since January 1. Which city has the greater average daily temperature? Explain.

Answer:

Question 25.

**USING TOOLS**

The table shows the numbers of employees N (in thousands) at a sporting goods company each year for 11 years. The time t is measured in years, with t = 1 representing the first year.

a. Use sinusoidal regression to find a model that gives N as a function of t.

b. Predict the number of employees at the company in the 12th year.

Answer:

Question 26.

**THOUGHT PROVOKING**

The figure shows a tangent line drawn to the graph of the function y = sin x. At several points on the graph, draw a tangent line to the graph and estimate its slope. Then plot the points (x, m), where m is the slope of the tangent line. What can you conclude?

Answer:

Question 27.

**REASONING**

Determine whether you would use a sine or cosine function to model each sinusoid with the y-intercept described. Explain your reasoning.

a. The y-intercept occurs at the maximum value of the function.

b. The y-intercept occurs at the minimum value of the function.

c. The y-intercept occurs halfway between the maximum and minimum values of the function.

Answer:

Question 28.

**HOW DO YOU SEE IT?**

What is the frequency of the function whose graph is shown? Explain.

Answer:

Question 29.

**USING STRUCTURE**

During one cycle, a sinusoid has a minimum at (\(\frac{\pi}{2}\), 3 ) and a maximum at (\(\frac{\pi}{4}\), 8 ). Write a sine function and a cosine function for the sinusoid. Use a graphing calculator to verify that your answers are correct.

Answer:

Question 30.

**MAKING AN ARGUMENT**

Your friend claims that a function with a frequency of 2 has a greater period than a function with a frequency of \(\frac{1}{2}\). Is your friend correct? Explain your reasoning.

Answer:

Question 31.

**PROBLEM SOLVING**

The low tide at a port is 3.5 feet and occurs at midnight. After 6 hours, the port is at high tide, which is 16.5 feet.

a. Write a sinusoidal model that gives the tide depth d(in feet) as a function of the time t(in hours). Let t = 0 represent midnight.

b. Find all the times when low and high tides occur in a 24-hour period.

c. Explain how the graph of the function you wrote in part (a) is related to a graph that shows the tide depth d at the port t hours after 3:00 A.M.

Answer:

**Maintaining Mathematical Proficiency**

**Simplify the expression.**

Question 32.

\(\frac{17}{\sqrt{2}}\)

Answer:

Question 33.

\(\frac{3}{\sqrt{6}-2}\)

Answer:

Question 34.

\(\frac{8}{\sqrt{10}+3}\)

Answer:

Question 35.

\(\frac{13}{\sqrt{3}+\sqrt{11}}\)

Answer:

**Expand the logarithmic expression.**

Question 36.

log_{8}\(\frac{x}{7}\)

Answer:

Question 37.

ln 2x

Answer:

Question 38.

log_{3} 5x^{3}

Answer:

Question 39.

ln \(\frac{4 x^{6}}{y}\)

Answer:

### Lesson 9.7 Using Trigonometric Identities

**Essential Question** How can you verify a trigonometric identity?

**EXPLORATION 1**

Writing a Trigonometric Identity

Work with a partner. In the figure, the point (x, y) is on a circle of radius c with center at the origin.

a. Write an equation that relates a, b, and c.

b. Write expressions for the sine and cosine ratios of angle θ.

c. Use the results from parts (a) and (b) to find the sum of sin^{2}θ and cos^{2}θ. What do you observe?

d. Complete the table to verify that the identity you wrote in part (c) is valid for angles (of your choice) in each of the four quadrants.

**EXPLORATION 2**

Writing Other Trigonometric Identities

Work with a partner. The trigonometric identity you derived in Exploration 1 is called a Pythagorean identity. There are two other Pythagorean identities. To derive them, recall the four relationships:

a. Divide each side of the Pythagorean identity you derived in Exploration 1 by cos^{2}θ and simplify. What do you observe?

b. Divide each side of the Pythagorean identity you derived in Exploration 1 by sin^{2}θ and simplify. What do you observe?

**Communicate Your Answer**

Question 3.

How can you verify a trigonometric identity?

Answer:

Question 4.

Is sin θ = cos θ a trigonometric identity? Explain your reasoning.

Answer:

Question 5.

Give some examples of trigonometric identities that are different than those in Explorations 1 and 2.

Answer:

**Monitoring Progress**

Question 1.

Given that cos θ = \(\frac{1}{6}\) and 0 < θ < \(\frac{\pi}{2}\), find the values of the other five trigonometric functions of θ.

Answer:

**Simplify the expression.**

Question 2.

sin x cot x sec x

Answer:

Question 3.

cos θ − cos θ sin^{2}θ

Answer:

Question 4.

\(\frac{\tan x \csc x}{\sec x}\)

Answer:

**Verify the identity.**

Question 5.

cot(−θ) =−cot θ

Answer:

Question 6.

csc^{2}x(1 − sin2x) = cot^{2}x

Answer:

Question 7.

cos x csc x tan x = 1

Answer:

Question 8.

(tan^{2}x + 1)(cos^{2}x− 1) = −tan^{2}x

Answer:

### Using Trigonometric Identities 9.7 Exercises

**Vocabulary and Core Concept Check**

Question 1.

**WRITING**

Describe the difference between a trigonometric identity and a trigonometric equation.

Answer:

Question 2.

**WRITING**

Explain how to use trigonometric identities to determine whether sec(−θ) = sec θ or sec(−θ) = −sec θ.

Answer:

**Monitoring Progress and Modeling with Mathematics**

**In Exercises 3–10, find the values of the other five trigonometric functions of θ.**

Question 3.

sinθ = \(\frac{1}{3}\), 0 < θ < \(\frac{\pi}{2}\)

Answer:

Question 4.

sin θ = −\(\frac{7}{10}\), π < θ < \(\frac{3 \pi}{2}\)

Answer:

Question 5.

tanθ = −\(\frac{3}{7}\), \(\frac{\pi}{2}\) < θ < π

Answer:

Question 6.

cot θ = −\(\frac{2}{5}\), \(\frac{\pi}{2}\) < θ < π

Answer:

Question 7.

cos θ = −\(\frac{5}{6}\), π < θ < \(\frac{3 \pi}{2}\)

Answer:

Question 8.

sec θ = \(\frac{9}{4}\), \(\frac{3 \pi}{2}\) < θ < 2π

Answer:

Question 9.

cot θ = −3, \(\frac{3 \pi}{2}\) < θ < 2π

Answer:

Question 10.

csc θ = −\(\frac{5}{3}\), π < θ < \(\frac{3 \pi}{2}\)

Answer:

**In Exercises 11–20, simplify the expression.**

Question 11.

sin x cot x

Answer:

Question 12.

cos θ (1 + tan^{2}θ)

Answer:

Question 13.

\(\frac{\sin (-\theta)}{\cos (-\theta)}\)

Answer:

Question 14.

\(\frac{\cos ^{2} x}{\cot ^{2} x}\)

Answer:

Question 15.

\(\frac{\cos \left(\frac{\pi}{2}-x\right)}{\csc x} \)

Answer:

Question 16.

sin(\(\frac{\pi}{2}\) – θ) sec θ

Answer:

Question 17.

\(\frac{\csc ^{2} x-\cot ^{2} x}{\sin (-x) \cot x}\)

Answer:

Question 18.

\(\frac{\cos ^{2} x \tan ^{2}(-x)-1}{\cos ^{2} x}\)

Answer:

Question 19.

\(\frac{\cos \left(\frac{\pi}{2}-\theta\right)}{\csc \theta}\) + cos^{2} θ

Answer:

Question 20.

\(\frac{\sec x \sin x+\cos \left(\frac{\pi}{2}-x\right)}{1+\sec x}\)

Answer:

**ERROR ANALYSIS** In Exercises 21 and 22, describe and correct the error in simplifying the expression.

Question 21.

Answer:

Question 22.

Answer:

**In Exercises 23–30, verify the identity.**

Question 23.

sin x csc x = 1

Answer:

Question 24.

tan θ csc θ cos θ = 1

Answer:

Question 25.

cos (\(\frac{3 \pi}{2}\) − x)cot x = cos x

Answer:

Question 26.

sin (\(\frac{3 \pi}{2}\) − x)tan x = sin x

Answer:

Question 27.

\(\frac{\cos \left(\frac{\pi}{2}-\theta\right)+1}{1-\sin (-\theta)}\) = 1

Answer:

Question 28.

\(\frac{\sin ^{2}(-x)}{\tan ^{2} x}\) = cos^{2} x

Answer:

Question 29.

\(\frac{1+\cos x}{\sin x}+\frac{\sin x}{1+\cos x}\) = 2 csc x

Answer:

Question 30.

\(\frac{\sin x}{1-\cos (-x)}\) = csc x + cot x

Answer:

Question 31.

**USING STRUCTURE**

A function f is odd when f(−x) = −f(x). A function f is even when (−x) = f (x). Which of the six trigonometric functions are odd? Which are even? Justify your answers using identities and graphs.

Answer:

Question 32.

**ANALYZING RELATIONSHIPS**

As the value of cos θ increases, what happens to the value of sec θ? Explain your reasoning.

Answer:

Question 33.

**MAKING AN ARGUMENT**

Your friend simplifies an expression and obtains sec x tan x− sin x. You simplify the same expression and obtain sin x tan2x. Are your answers equivalent? Justify your answer.

Answer:

Question 34.

**HOW DO YOU SEE IT?**

The figure shows the unit circle and the angle θ.

a. Is sin θ positive or negative? cos θ? tan θ?

b. In what quadrant does the terminal side of −θ lie?

c. Is sin(−θ) positive or negative? cos(−θ)? tan(−θ)?

Answer:

Question 35.

**MODELING WITH MATHEMATICS**

A vertical gnomon(the part of a sundial that projects a shadow) has height h. The length s of the shadow cast by the gnomon when the angle of the Sun above the horizon is θ can be modeled by the equation below. Show that the equation below is equivalent to s = h cot θ.

Answer:

Question 36.

**THOUGHT PROVOKING**

Explain how you can use a trigonometric identity to find all the values of x for which sin x = cos x.

Answer:

Question 37.

**DRAWING CONCLUSIONS**

Static friction is the amount of force necessary to keep a stationary object on a flat surface from moving. Suppose a book weighing W pounds is lying on a ramp inclined at an angle θ. The coefficient of static friction u for the book can be found using the equation uW cos θ = W sin θ.

a. Solve the equation for u and simplify the result.

b. Use the equation from part (a) to determine what happens to the value of u as the angle θ increases from 0° to 90°.

Answer:

Question 38.

**PROBLEM SOLVING**

When light traveling in a medium (such as air) strikes the surface of a second medium (such as water) at an angle θ_{1}, the light begins to travel at a different angle θ_{2}. This change of direction is defined by Snell’s law, n_{1} sin θ_{1} = n_{2} sin θ_{2}, where n_{1} and n_{2} are the indices of refraction for the two mediums. Snell’s law can be derived from the equation.

a. Simplify the equation to derive Snell’s law.

b. What is the value of n1 when θ_{1} = 55°, θ_{2} = 35°, and n_{2} = 2?

c. If θ_{1} = θ_{2}, then what must be true about the values of n_{1} and n_{2}? Explain when this situation would occur.

Answer:

Question 39.

**WRITING**

Explain how transformations of the graph of the parent function f(x) = sin x support the cofunction identity sin (\(\frac{\pi}{2}\) − θ) = cos θ.

Answer:

Question 40.

**USING STRUCTURE**

Verify each identity.

a. ln ∣sec θ∣= −ln ∣cos θ∣

b. ln ∣tan θ∣= ln ∣sin θ∣− ln ∣cos θ∣

Answer:

**Maintaining Mathematical Proficiency**

**Find the value of x for the right triangle.**

Question 41.

Answer:

Question 42.

Answer:

Question 43.

Answer:

### Lesson 9.8 Using Sum and Difference Formulas

**Essential Question** How can you evaluate trigonometric functions of the sum or difference of two angles?

**EXPLORATION 1**

Deriving a Difference Formula

Work with a partner.

a. Explain why the two triangles shown are congruent.

b. Use the Distance Formula to write an expression for d in the first unit circle.

c. Use the Distance Formula to write an expression for d in the second unit circle.

d. Write an equation that relates the expressions in parts (b) and (c). Then simplify this equation to obtain a formula for cos(a − b).

**EXPLORATION 2**

Deriving a Sum Formula

Work with a partner. Use the difference formula you derived in Exploration 1 to write a formula for cos(a+b) in terms of sine and cosine of a and b. Hint: Use the fact that cos(a+b) = cos[a − (−b)].

**EXPLORATION 3**

Deriving Difference and Sum Formulas

Work with a partner. Use the formulas you derived in Explorations 1 and 2 to write formulas for sin(a − b) and sin(a + b) in terms of sine and cosine of a and b. Hint: Use the cofunction identities sin (\(\frac{\pi}{2}\) − a)= cos a and cos (\(\frac{\pi}{2}\) − a)= sin a and the fact that

cos [(\(\frac{\pi}{2}\) − a) + b ]= sin(a − b) and sin(a+b) = sin[a − (−b)].

**Communicate Your Answer**

Question 4.

How can you evaluate trigonometric functions of the sum or difference of two angles?

Answer:

Question 5.

a. Find the exact values of sin 75° and cos 75° using sum formulas. Explain your reasoning.

b. Find the exact values of sin 75° and cos 75° using difference formulas. Compare your answers to those in part (a).

Answer:

**Monitoring Progress**

**Find the exact value of the expression.**

Question 1.

sin 105°

Answer:

Question 2.

cos 15°

Answer:

Question 3.

tan \(\frac{5 \pi}{2}\)

Answer:

Question 4.

cos \(\frac{\pi}{12}\)

Answer:

Question 5.

Find sin(a−b) given that sin a = \(\frac{8}{17}\) with 0 < a < \(\frac{\pi}{2}\) and cos b = −\(\frac{24}{25}\) with π < b < \(\frac{3 \pi}{2}\) .

Answer:

**Simplify the expression.**

Question 6.

sin(x + π)

Answer:

Question 7.

cos(x − 2π)

Answer:

Question 8.

tan(x − π)

Answer:

Question 9.

Solve sin (\(\frac{\pi}{4}\) − x)− sin (x + \(\frac{\pi}{4}\))= 1 for 0 ≤ x < 2π.

Answer:

### Using Sum and Difference Formulas 9.8 Exercises

**Vocabulary and Core Concept Check**

Question 1.

**COMPLETE THE SENTENCE**

Write the expression cos 130° cos 40°− sin 130° sin 40° as the cosine of an angle.

Answer:

Question 2.

**WRITING**

Explain how to evaluate tan 75° using either the sum or difference formula for tangent.

Answer:

**Monitoring Progress and Modeling with Mathematics**

**In Exercises 3–10, find the exact value of the expression.**

Question 3.

tan(−15°)

Answer:

Question 4.

tan 195°

Answer:

Question 5.

sin \(\frac{23 \pi}{12}\)

Answer:

Question 6.

sin(−165°)

Answer:

Question 7.

cos 105°

Answer:

Question 8.

cos \(\frac{11 \pi}{12}\)

Answer:

Question 9.

tan \(\frac{17 \pi}{12}\)

Answer:

Question 10.

sin (−\(\frac{7 \pi}{12}\))

Answer:

In Exercises 11–16, evaluate the expression given that cos a = \(\frac{4}{5}\) with 0 < a < \(\frac{\pi}{2}\) and sin b = –\(\frac{15}{17}\) with \(\frac{3 \pi}{2}\) < b < 2π.

Question 11.

sin(a + b)

Answer:

Question 12.

sin(a − b)

Answer:

Question 13.

cos(a − b)

Answer:

Question 14.

cos(a + b)

Answer:

Question 15.

tan(a + b)

Answer:

Question 16.

tan(a − b)

Answer:

**In Exercises 17–22, simplify the expression.**

Question 17.

tan(x + π)

Answer:

Question 18.

cos (x − \(\frac{\pi}{2}\))

Answer:

Question 19.

cos(x + 2π)

Answer:

Question 20.

tan(x − 2π)

Answer:

Question 21.

sin (x − \(\frac{3 \pi}{2}\))

Answer:

Question 22.

tan (x + \(\frac{\pi}{2}\))

Answer:

**ERROR ANALYSIS** In Exercises 23 and 24, describe and correct the error in simplifying the expression.

Question 23.

Answer:

Question 24.

Answer:

Question 25.

What are the solutions of the equation 2 sin x − 1 = 0 for 0 ≤ x < 2π?

A. \(\frac{\pi}{3}\)

B. \(\frac{\pi}{6}\)

C. \(\frac{2 \pi}{3}\)

D. \(\frac{5 \pi}{6}\)

Answer:

Question 26.

What are the solutions of the equation tan x + 1 = 0 for 0 ≤ x < 2π?

A. \(\frac{\pi}{4}\)

B. \(\frac{3 \pi}{4}\)

C. \(\frac{5 \pi}{4}\)

D. \(\frac{7 \pi}{4}\)

Answer:

**In Exercises 27–32, solve the equation for 0 ≤ x < 2π.**

Question 27.

sin (x + \(\frac{\pi}{2}\)) = \(\frac{1}{2}\)

Answer:

Question 28.

tan (x − \(\frac{\pi}{4}\)) = 0

Answer:

Question 29.

cos (x +\(\frac{\pi}{6}\)) − cos (x −\(\frac{\pi}{6}\)) = 1

Answer:

Question 30.

sin (x + \(\frac{\pi}{4}\)) + sin (x − \(\frac{\pi}{4}\)) = 0

Answer:

Question 31.

tan(x + π) − tan(π − x) = 0

Answer:

Question 32.

sin(x + π) + cos(x + π) = 0

Answer:

Question 33.

**USING EQUATIONS**

Derive the cofunction identity sin (\(\frac{\pi}{2}\) − θ)= cos θ using the difference formula for sine.

Answer:

Question 34.

**MAKING AN ARGUMENT**

Your friend claims it is possible to use the difference formula for tangent to derive the cofunction identity tan (\(\frac{\pi}{2}\) − θ) = cot θ. Is your friend correct? Explain your reasoning.

Answer:

Question 35.

**MODELING WITH MATHEMATICS**

A photographer is at a height h taking aerial photographs with a 35-millimeter camera. The ratio of the image length WQ to the length NA of the actual object is given by the formula

where θ is the angle between the vertical line perpendicular to the ground and the line from the camera to point A and t is the tilt angle of the film. When t = 45°, show that the formula can be rewritten as \(\frac{W Q}{N A}=\frac{70}{h(1+\tan \theta)}\).

Answer:

Question 36.

**MODELING WITH MATHEMATICS**

When a wave travels through a taut string, the displacement y of each point on the string depends on the time t and the point’s position x. The equation of a standing wave can be obtained by adding the displacements of two waves traveling in opposite directions. Suppose a standing wave can be modeled by the formula

y = A cos (\(\frac{2 \pi t}{3}-\frac{2 \pi x}{5}\)) + A cos (\(\frac{2 \pi t}{3}+\frac{2 \pi x}{5}\)) .When t= 1, show that the formula can be rewritten as y = −A cos \(\frac{2 \pi x}{5}\).

Answer:

Question 37.

**MODELING WITH MATHEMATICS**

The busy signal on a touch-tone phone is a combination of two tones with frequencies of 480 hertz and 620 hertz. The individual tones can be modeled by the equations:

480 hertz: y_{1} = cos 960πt

620 hertz: y_{2} = cos 1240πt

The sound of the busy signal can be modeled by y_{1} + y_{2}. Show that y_{1} + y_{2} = 2 cos 1100πt cos 140πt.

Answer:

Question 38.

**HOW DO YOU SEE IT?**

Explain how to use the figure to solve the equation sin (x + \(\frac{\pi}{4}\)) − sin (\(\frac{\pi}{4}\) − x) = 0 for 0 ≤ x < 2π.

Answer:

Question 39.

**MATHEMATICAL CONNECTIONS**

The figure shows the acute angle of intersection, θ_{2} − θ_{1}, of two lines with slopes m_{1} and m_{2}.

a. Use the difference formula for tangent to write an equation for tan (θ_{2} − θ_{1}) in terms of m_{1} and m_{2}.

b. Use the equation from part (a) to find the acute angle of intersection of the lines y = x− 1 and y = \(\left(\frac{1}{\sqrt{3}-2}\right)\)x + \(\frac{4-\sqrt{3}}{2-\sqrt{3}}\).

Answer:

Question 40.

**THOUGHT PROVOKING**

Rewrite each function. Justify your answers.

a. Write sin 3x as a function of sin x.

b. Write cos 3x as a function of cos x.

c. Write tan 3x as a function of tan x.

Answer:

**Maintaining Mathematical Proficiency**

**Solve the equation. Check your solution(s).**

Question 41.

1 − \(\frac{9}{x-2}\) = −\(\frac{7}{2}\)

Answer:

Question 42.

\(\frac{12}{x}\) + \(\frac{3}{4}\) = \(\frac{8}{x}\)

Answer:

Question 43.

\(\frac{2 x-3}{x+1}\) = \(\frac{10}{x^{2}-1}\) + 5

Answer:

### Trigonometric Ratios and Functions Performance Task: Lightening the Load

**9.5–9.8 What Did You Learn?**

**Core Vocabulary**

frequency, p. 506

sinusoid, p. 507

trigonometric identity, p. 514

**Core Concepts**

Section 9.5

Characteristics of y = tan x and y = cot x, p. 498

Period and Vertical Asymptotes of y = a tan bx and y = a cot bx, p. 499

Characteristics of y = sec x and y = csc x, p. 500

Section 9.6

Frequency, p. 506

Writing Trigonometric Functions, p. 507

Using Technology to Find Trigonometric Models, p. 509

Section 9.7

Fundamental Trigonometric Identities, p. 514

Section 9.8

Sum and Difference Formulas, p. 520

Trigonometric Equations and Real-Life Formulas, p. 522

**Mathematical Practices**

Question 1.

Explain why the relationship between θ and d makes sense in the context of the situation in Exercise 43 on page 503.

Answer:

Question 2.

How can you use definitions to relate the slope of a line with the tangent of an angle in Exercise 39 on page 524?

Answer:

**Performance Task: Lightening the Load**

You need to move a heavy table across the room. What is the easiest way to move it? Should you push it? Should you tie a rope around one leg of the table and pull it? How can trigonometry help you make the right decision?

To explore the answers to these questions and more, go to BigIdeasMath.com.

### Trigonometric Ratios and Functions Chapter Review

**9.1 Right Triangle Trigonometry (pp. 461−468)**

Question 1.

In a right triangle, θ is an acute angle and cos θ = \(\frac{6}{11}\). Evaluate the other five trigonometric functions of θ.

Answer:

Question 2.

The shadow of a tree measures 25 feet from its base. The angle of elevation to the Sun is 31°. How tall is the tree?

Answer:

**9.2 Angles and Radian Measure (pp. 469−476)**

Question 3.

Find one positive angle and one negative angle that are coterminal with 382°.

Answer:

Convert the degree measure to radians or the radian measure to degrees.

Question 4.

30°

Answer:

Question 5.

225°

Answer:

Question 6.

\(\frac{3 \pi}{4}\)

Answer:

Question 7.

\(\frac{5 \pi}{3}\)

Answer:

Question 8.

A sprinkler system on a farm rotates 140°and sprays water up to 35 meters. Draw a diagram that shows the region that can be irrigated with the sprinkler. Then find the area of the region.

Answer:

**9.3 Trigonometric Functions of Any Angle (pp. 477−484)**

**Evaluate the six trigonometric functions of θ.**

Question 9.

Answer:

Question 10.

Answer:

Question 11.

Answer:

**Evaluate the function without using a calculator.**

Question 12.

tan 330°

Answer:

Question 13.

sec(−405°)

Answer:

Question 14.

sin \(\frac{13 \pi}{6}\)

Answer:

Question 15.

sec \(\frac{11 \pi}{3}\)

Answer:

**9.4 Graphing Sine and Cosine Functions (pp. 485−494)**

Identify the amplitude and period of the function. Then graph the function and describe the graph of g as a transformation of the graph of the parent function.

Question 16.

g(x) = 8 cos x

Answer:

Question 17.

g(x) = 6 sin πx

Answer:

Question 18.

g(x) = \(\frac{1}{4}\) cos 4x

Answer:

Graph the function.

Question 19.

g(x) = cos(x + π) + 2

Answer:

Question 20.

g(x) = −sin x − 4

Answer:

Question 21.

g(x) = 2 sin (x + \(\frac{\pi}{2}\))

Answer:

**9.5 Graphing Other Trigonometric Functions (pp. 497−504)**

Graph one period of the function. Describe the graph of g as a transformation of the graph of its parent function.

Question 22.

g(x) = tan \(\frac{1}{2}\)x

Answer:

Question 23.

g(x) = 2 cot x

Answer:

Question 24.

g(x) = 4 tan 3πx

Answer:

Graph the function.

Question 25.

g(x) = 5 csc x

Answer:

Question 26.

g(x) = sec \(\frac{1}{2}\)x

Answer:

Question 27.

g(x) = 5 sec πx

Answer:

Question 28.

g(x) = \(\frac{1}{2}\) csc \(\frac{\pi}{4}\)x

Answer:

**9.6 Modeling with Trigonometric Functions (pp. 505−512)**

**Write a function for the sinusoid.**

Question 29.

Answer:

Question 30.

Answer:

Question 31.

You put a reflector on a spoke of your bicycle wheel. The highest point of the reflector is 25 inches above the ground, and the lowest point is 2 inches. The reflector makes 1 revolution per second. Write a model for the height h (in inches) of a reflector as a function of time t (in seconds) given that the reflector is at its lowest point when t = 0.

Answer:

Question 32.

The table shows the monthly precipitation P (in inches) for Bismarck, North Dakota, where t = 1 represents January. Write a model that gives P as a function of t and interpret the period of its graph.

Answer:

**9.7 Using Trigonometric Identities (pp. 513−518)**

**Simplify the expression.**

Question 33.

cot^{2}x − cot^{2}x cos^{2}x

Answer:

Question 34.

\(\frac{(\sec x+1)(\sec x-1)}{\tan x}\)

Answer:

Question 35.

sin (\(\frac{\pi}{2}\) − x)tan x

Answer:

**Verify the identity.**

Question 36.

\(\frac{\cos x \sec x}{1+\tan ^{2} x}\) = cos^{2}x

Answer:

Question 37.

tan (\(\frac{\pi}{2}\) − x)cot x = csc^{2}x − 1

Answer:

**9.8 Using Sum and Difference Formulas (pp. 519−524)**

**Find the exact value of the expression.**

Question 38.

sin 75°

Answer:

Question 39.

tan(−15°)

Answer:

Question 40.

cos \(\frac{\pi}{12}\)

Answer:

Question 41.

Find tan(a + b), given that tan a = \(\frac{1}{4}\) with π < a < \(\frac{3 \pi}{2}\) and tan b = \(\frac{3}{7}\) with 0 < b < \(\frac{\pi}{2}\) .

Answer:

Solve the equation for 0 ≤ x < 2π.

Question 42.

cos (x + \(\frac{3 \pi}{4}\)) + cos (x − \(\frac{3 \pi}{4}\)) = 1

Answer:

Question 43.

tan(x + π) + cos (x + \(\frac{\pi}{2}\))= 0

Answer:

### Trigonometric Ratios and Functions Chapter Test

**Verify the identity.**

Question 1.

\(\frac{\cos ^{2} x+\sin ^{2} x}{1+\tan ^{2} x}\) = cos^{2}x

Answer:

Question 2.

\(\frac{1+\sin x}{\cos x}+\frac{\cos x}{1+\sin x}\) = 2 sec x

Answer:

Question 3.

cos (x + \(\frac{3 \pi}{2}\)) = sin x

Answer:

Question 4.

Evaluate sec(−300°) without using a calculator.

Answer:

Write a function for the sinusoid.

Question 5.

Answer:

Question 6.

Answer:

Graph the function. Then describe the graph of g as a transformation of the graph of its parent function.

Question 7.

g(x) = −4 tan 2x

Answer:

Question 8.

g(x) = −2 cos \(\frac{1}{3}\)x + 3

Answer:

Question 9.

g(x) = 3 csc πx

Answer:

Convert the degree measure to radians or the radian measure to degrees. Then find one positive angle and one negative angle that are coterminal with the given angle.

Question 10.

−50°

Answer:

Question 11.

\(\frac{4 \pi}{5}\)

Answer:

Question 12.

\(\frac{8 \pi}{3}\)

Answer:

Question 13.

Find the arc length and area of a sector with radius r = 13 inches and central angle θ = 40°.

Answer:

Evaluate the six trigonometric functions of the angle θ.

Question 14.

Answer:

Question 15.

Answer:

Question 16.

In which quadrant does the terminal side of θ lie when cos θ < 0 and tan θ > 0? Explain.

Answer:

Question 17.

How tall is the building? Justify your answer.

Answer:

Question 18.

The table shows the average daily high temperatures T (in degrees Fahrenheit) in Baltimore, Maryland, where m= 1 represents January. Write a model that gives T as a function of m and interpret the period of its graph.

Answer:

### Trigonometric Ratios and Functions Cumulative Assessment

Question 1.

Which expressions are equivalent to 1?

Answer:

Question 2.

Which rational expression represents the ratio of the perimeter to the area of the playground shown in the diagram?

Answer:

Question 3.

The chart shows the average monthly temperatures (in degrees Fahrenheit) and the gas usages (in cubic feet) of a household for 12 months.

a. Use a graphing calculator to find trigonometric models for the average temperature y_{1} as a function of time and the gas usage y_{2} (in thousands of cubic feet) as a function of time. Let t = 1 represent January.

b. Graph the two regression equations in the same coordinate plane on your graphing calculator. Describe the relationship between the graphs.

Answer:

Question 4.

Evaluate each logarithm using log_{2} 5 ≈ 2.322 and log_{2} 3 ≈ 1.585, if necessary. Then order the logarithms by value from least to greatest.

a. log 1000

b. log_{2} 15

c. ln e

d. log_{2} 9

e. log_{2}\(\frac{5}{3}\)

f. log_{2} 1

Answer:

Question 5.

Which function is not represented by the graph?

Answer:

Question 6.

Complete each statement with < or > so that each statement is true.

Answer:

Question 7.

Use the Rational Root Theorem and the graph to find all the real zeros of the function f(x) = 2x^{3} − x^{2} − 13x− 6. (HSA-APR.B.3)

Answer:

Question 8.

Your friend claims −210° is coterminal with the angle \(\frac{5 \pi}{6}\). Is your friend correct? Explain your reasoning.

Answer:

Question 9.

Company A and Company B offer the same starting annual salary of $20,000. Company A gives a $1000 raise each year. Company B gives a 4% raise each year.

a. Write rules giving the salaries an and bn for your nth year of employment at Company A and Company B, respectively. Tell whether the sequence represented by each rule is arithmetic, geometric, or neither.

b. Graph each sequence in the same coordinate plane.

c. Under what conditions would you choose to work for Company B?

d. After 20 years of employment, compare your total earnings.

Answer: