13- 4 Inverses of Trigonometric Functions 953

13-4 KEYWORD: MB7 13-4

GUIDED PRACTICE 1. Vocabulary Explain how the inverse tangent function differs from the reciprocal

of the tangent function.

SEE EXAMPLE 1 p. 950

Find all possible values of each expression.

2. sin -1 (- 1 _ 2

) 3. tan -1 √ " 3 _ 3

4. co s -1 (- √ " 2 _ 2

)

SEE EXAMPLE 2 p. 951

Evaluate each inverse trigonometric function. Give your answer in both radians and degrees.

5. Cos -1 √ " 3 _ 2

6. Tan -1 1 7. Cos -1 2

8. Tan -1 (- √ " 3 ) 9. Sin -1 √ " 2 _ 2

10. Sin -1 0

SEE EXAMPLE 3 p. 952

11. Architecture A point on the top of the Leaning Tower of Pisa is shifted about 13.5 ft horizontally compared with the tower’s base. To the nearest degree, how many degrees does the tower tilt from vertical?

SEE EXAMPLE 4 p. 952

Solve each equation to the nearest tenth. Use the given restrictions.

12. tan θ = 1.4, for -90° < θ < 90°

13. tan θ = 1.4, for 180° < θ < 270°

14. cos θ = -0.25, for 0 ≤ θ ≤ 180°

15. cos θ = -0.25, for 180° < θ < 270°

KEYWORD: MB7 Parent

THINK AND DISCUSS 1. Given that θ is an acute angle in a right triangle, describe the measurements

that you need to know to find the value of θ by using the inverse cosine function.

2. Explain the difference between tan -1 a and Tan -1 a.

3. GET ORGANIZED Copy and complete the graphic organizer. In each box, give the indicated property of the inverse trigonometric functions.

ExercisesExercises

a207se_c13l04_0950_0955.indd 953a207se_c13l04_0950_0955.indd 953 11/21/05 2:47:00 PM11/21/05 2:47:00 PM

             

954 Chapter 13 Trigonometric Functions

A flight simulator is a device used in training pilots that mimics flight conditions as realistically as possible. Some flight simulators involve full-size cockpits equipped with sound, visual, and motion systems.

Aviation

PRACTICE AND PROBLEM SOLVING

For See Exercises Example

16–18 1 19–24 2 25 3 26–29 4

Independent Practice Find all possible values of each expression.

16. cos -1 1 17. sin -1 √ " 3 _ 2

18. tan -1 (-1)

Evaluate each inverse trigonometric function. Give your answer in both radians and degrees.

19. Sin -1 √ " 3 _ 2

20. Cos -1 (-1) 21. Tan -1 (- √ " 3 _ 3

)

22. Cos -1 (- √ " 3 _ 2

) 23. Tan -1 √ " 3 24. Sin -1 √ " 3

25. Volleyball A volleyball player spikes the ball from a height of 2.44 m. Assume that the path of the ball is a straight line. To the nearest degree, what is the maximum angle θ at which the ball can be hit and land within the court?

Solve each equation to the nearest tenth. Use the given restrictions.

26. sin θ = -0.75, for -90° ≤ θ ≤ 90° 27. sin θ = -0.75, for 180° < θ < 270°

28. cos θ = 0.1, for 0° ≤ θ ≤ 180° 29. cos θ = 0.1, for 270° < θ < 360°

30. Aviation The pilot of a small plane is flying at an altitude of 2000 ft. The pilot plans to start the final descent toward a runway when the horizontal distance between the plane and the runway is 2 mi. To the nearest degree, what will be the angle of depression θ from the plane to the runway at this point?

31. Multi-Step The table shows the

Pool Style

Length(ft)

Shallow End Depth

(ft)Deep End Depth (ft)

A 38 3 8

B 25 2 6

C 50 2.5 7

dimensions of three pool styles offered by a construction company.a. To the nearest tenth of a degree,

what angle θ does the bottom of each pool make with the horizontal?

b. Which pool style’s bottom has the steepest slope? Explain.

c. What if...? If the slope of the bottom of a pool can be no greater than 1 __ 6 , what is the greatest angle θ that the bottom of the pool can make with the horizontal? Round to the nearest tenth of a degree.

32. Navigation Lines of longitude are closer together near the poles than at the equator. The formula for the length " of 1° of longitude in miles is " = 69.0933 cos θ, where θ is the latitude in degrees.

a. At what latitude, to the nearest degree, is the length of a degree of longitude approximately 59.8 miles?

b. To the nearest mile, how much longer is the length of a degree of longitude at the equator, which has a latitude of 0°, than at the Arctic Circle, which has a latitude of about 66°N?

Skills Practice p. S29Application Practice p. S44

Extra Practice

a207se_c13l04_0950_0955.indd 954a207se_c13l04_0950_0955.indd 954 11/21/05 2:47:05 PM11/21/05 2:47:05 PM

         

954 Chapter 13 Trigonometric Functions

A flight simulator is a device used in training pilots that mimics flight conditions as realistically as possible. Some flight simulators involve full-size cockpits equipped with sound, visual, and motion systems.

Aviation

PRACTICE AND PROBLEM SOLVING

For See Exercises Example

16–18 1 19–24 2 25 3 26–29 4

Independent Practice Find all possible values of each expression.

16. cos -1 1 17. sin -1 √ " 3 _ 2

18. tan -1 (-1)

Evaluate each inverse trigonometric function. Give your answer in both radians and degrees.

19. Sin -1 √ " 3 _ 2

20. Cos -1 (-1) 21. Tan -1 (- √ " 3 _ 3

)

22. Cos -1 (- √ " 3 _ 2

) 23. Tan -1 √ " 3 24. Sin -1 √ " 3

25. Volleyball A volleyball player spikes the ball from a height of 2.44 m. Assume that the path of the ball is a straight line. To the nearest degree, what is the maximum angle θ at which the ball can be hit and land within the court?

Solve each equation to the nearest tenth. Use the given restrictions.

26. sin θ = -0.75, for -90° ≤ θ ≤ 90° 27. sin θ = -0.75, for 180° < θ < 270°

28. cos θ = 0.1, for 0° ≤ θ ≤ 180° 29. cos θ = 0.1, for 270° < θ < 360°

30. Aviation The pilot of a small plane is flying at an altitude of 2000 ft. The pilot plans to start the final descent toward a runway when the horizontal distance between the plane and the runway is 2 mi. To the nearest degree, what will be the angle of depression θ from the plane to the runway at this point?

31. Multi-Step The table shows the

Pool Style

Length(ft)

Shallow End Depth

(ft)Deep End Depth (ft)

A 38 3 8

B 25 2 6

C 50 2.5 7

dimensions of three pool styles offered by a construction company.a. To the nearest tenth of a degree,

what angle θ does the bottom of each pool make with the horizontal?

b. Which pool style’s bottom has the steepest slope? Explain.

c. What if...? If the slope of the bottom of a pool can be no greater than 1 __ 6 , what is the greatest angle θ that the bottom of the pool can make with the horizontal? Round to the nearest tenth of a degree.

32. Navigation Lines of longitude are closer together near the poles than at the equator. The formula for the length " of 1° of longitude in miles is " = 69.0933 cos θ, where θ is the latitude in degrees.

a. At what latitude, to the nearest degree, is the length of a degree of longitude approximately 59.8 miles?

b. To the nearest mile, how much longer is the length of a degree of longitude at the equator, which has a latitude of 0°, than at the Arctic Circle, which has a latitude of about 66°N?

Skills Practice p. S29Application Practice p. S44

Extra Practice

a207se_c13l04_0950_0955.indd 954a207se_c13l04_0950_0955.indd 954 11/21/05 2:47:05 PM11/21/05 2:47:05 PM

           

954 Chapter 13 Trigonometric Functions

A flight simulator is a device used in training pilots that mimics flight conditions as realistically as possible. Some flight simulators involve full-size cockpits equipped with sound, visual, and motion systems.

Aviation

PRACTICE AND PROBLEM SOLVING

For See Exercises Example

16–18 1 19–24 2 25 3 26–29 4

Independent Practice Find all possible values of each expression.

16. cos -1 1 17. sin -1 √ " 3 _ 2

18. tan -1 (-1)

Evaluate each inverse trigonometric function. Give your answer in both radians and degrees.

19. Sin -1 √ " 3 _ 2

20. Cos -1 (-1) 21. Tan -1 (- √ " 3 _ 3

)

22. Cos -1 (- √ " 3 _ 2

) 23. Tan -1 √ " 3 24. Sin -1 √ " 3

25. Volleyball A volleyball player spikes the ball from a height of 2.44 m. Assume that the path of the ball is a straight line. To the nearest degree, what is the maximum angle θ at which the ball can be hit and land within the court?

Solve each equation to the nearest tenth. Use the given restrictions.

26. sin θ = -0.75, for -90° ≤ θ ≤ 90° 27. sin θ = -0.75, for 180° < θ < 270°

28. cos θ = 0.1, for 0° ≤ θ ≤ 180° 29. cos θ = 0.1, for 270° < θ < 360°

30. Aviation The pilot of a small plane is flying at an altitude of 2000 ft. The pilot plans to start the final descent toward a runway when the horizontal distance between the plane and the runway is 2 mi. To the nearest degree, what will be the angle of depression θ from the plane to the runway at this point?

31. Multi-Step The table shows the

Pool Style

Length(ft)

Shallow End Depth

(ft)Deep End Depth (ft)

A 38 3 8

B 25 2 6

C 50 2.5 7

dimensions of three pool styles offered by a construction company.a. To the nearest tenth of a degree,

what angle θ does the bottom of each pool make with the horizontal?

b. Which pool style’s bottom has the steepest slope? Explain.

c. What if...? If the slope of the bottom of a pool can be no greater than 1 __ 6 , what is the greatest angle θ that the bottom of the pool can make with the horizontal? Round to the nearest tenth of a degree.

32. Navigation Lines of longitude are closer together near the poles than at the equator. The formula for the length " of 1° of longitude in miles is " = 69.0933 cos θ, where θ is the latitude in degrees.

a. At what latitude, to the nearest degree, is the length of a degree of longitude approximately 59.8 miles?

b. To the nearest mile, how much longer is the length of a degree of longitude at the equator, which has a latitude of 0°, than at the Arctic Circle, which has a latitude of about 66°N?

Skills Practice p. S29Application Practice p. S44

Extra Practice

a207se_c13l04_0950_0955.indd 954a207se_c13l04_0950_0955.indd 954 11/21/05 2:47:05 PM11/21/05 2:47:05 PM

           

954 Chapter 13 Trigonometric Functions

A flight simulator is a device used in training pilots that mimics flight conditions as realistically as possible. Some flight simulators involve full-size cockpits equipped with sound, visual, and motion systems.

Aviation

PRACTICE AND PROBLEM SOLVING

For See Exercises Example

16–18 1 19–24 2 25 3 26–29 4

Independent Practice Find all possible values of each expression.

16. cos -1 1 17. sin -1 √ " 3 _ 2

18. tan -1 (-1)

Evaluate each inverse trigonometric function. Give your answer in both radians and degrees.

19. Sin -1 √ " 3 _ 2

20. Cos -1 (-1) 21. Tan -1 (- √ " 3 _ 3

)

22. Cos -1 (- √ " 3 _ 2

) 23. Tan -1 √ " 3 24. Sin -1 √ " 3

25. Volleyball A volleyball player spikes the ball from a height of 2.44 m. Assume that the path of the ball is a straight line. To the nearest degree, what is the maximum angle θ at which the ball can be hit and land within the court?

Solve each equation to the nearest tenth. Use the given restrictions.

26. sin θ = -0.75, for -90° ≤ θ ≤ 90° 27. sin θ = -0.75, for 180° < θ < 270°

28. cos θ = 0.1, for 0° ≤ θ ≤ 180° 29. cos θ = 0.1, for 270° < θ < 360°

30. Aviation The pilot of a small plane is flying at an altitude of 2000 ft. The pilot plans to start the final descent toward a runway when the horizontal distance between the plane and the runway is 2 mi. To the nearest degree, what will be the angle of depression θ from the plane to the runway at this point?

31. Multi-Step The table shows the

Pool Style

Length(ft)

Shallow End Depth

(ft)Deep End Depth (ft)

A 38 3 8

B 25 2 6

C 50 2.5 7

dimensions of three pool styles offered by a construction company.a. To the nearest tenth of a degree,

what angle θ does the bottom of each pool make with the horizontal?

b. Which pool style’s bottom has the steepest slope? Explain.

c. What if...? If the slope of the bottom of a pool can be no greater than 1 __ 6 , what is the greatest angle θ that the bottom of the pool can make with the horizontal? Round to the nearest tenth of a degree.

32. Navigation Lines of longitude are closer together near the poles than at the equator. The formula for the length " of 1° of longitude in miles is " = 69.0933 cos θ, where θ is the latitude in degrees.

a. At what latitude, to the nearest degree, is the length of a degree of longitude approximately 59.8 miles?

b. To the nearest mile, how much longer is the length of a degree of longitude at the equator, which has a latitude of 0°, than at the Arctic Circle, which has a latitude of about 66°N?

Skills Practice p. S29Application Practice p. S44

Extra Practice

a207se_c13l04_0950_0955.indd 954a207se_c13l04_0950_0955.indd 954 11/21/05 2:47:05 PM11/21/05 2:47:05 PM

             

13- 4 Inverses of Trigonometric Functions 953

13-4 KEYWORD: MB7 13-4

GUIDED PRACTICE 1. Vocabulary Explain how the inverse tangent function differs from the reciprocal

of the tangent function.

SEE EXAMPLE 1 p. 950

Find all possible values of each expression.

2. sin -1 (- 1 _ 2

) 3. tan -1 √ " 3 _ 3

4. co s -1 (- √ " 2 _ 2

)

SEE EXAMPLE 2 p. 951

Evaluate each inverse trigonometric function. Give your answer in both radians and degrees.

5. Cos -1 √ " 3 _ 2

6. Tan -1 1 7. Cos -1 2

8. Tan -1 (- √ " 3 ) 9. Sin -1 √ " 2 _ 2

10. Sin -1 0

SEE EXAMPLE 3 p. 952

11. Architecture A point on the top of the Leaning Tower of Pisa is shifted about 13.5 ft horizontally compared with the tower’s base. To the nearest degree, how many degrees does the tower tilt from vertical?

SEE EXAMPLE 4 p. 952

Solve each equation to the nearest tenth. Use the given restrictions.

12. tan θ = 1.4, for -90° < θ < 90°

13. tan θ = 1.4, for 180° < θ < 270°

14. cos θ = -0.25, for 0 ≤ θ ≤ 180°

15. cos θ = -0.25, for 180° < θ < 270°

KEYWORD: MB7 Parent

THINK AND DISCUSS 1. Given that θ is an acute angle in a right triangle, describe the measurements

that you need to know to find the value of θ by using the inverse cosine function.

2. Explain the difference between tan -1 a and Tan -1 a.

3. GET ORGANIZED Copy and complete the graphic organizer. In each box, give the indicated property of the inverse trigonometric functions.

ExercisesExercises

a207se_c13l04_0950_0955.indd 953a207se_c13l04_0950_0955.indd 953 11/21/05 2:47:00 PM11/21/05 2:47:00 PM

NAME:  __________________________________      

DATE:_____/_____/_____                            HR:  _____  

ALG  2B  –  LESSON  13:4  -­‐    “  INVERSES  OF  TRIGONOMETRIC  FUNCTIONS”  

 SCORE:      _______    /16  

 

954 Chapter 13 Trigonometric Functions

A flight simulator is a device used in training pilots that mimics flight conditions as realistically as possible. Some flight simulators involve full-size cockpits equipped with sound, visual, and motion systems.

Aviation

PRACTICE AND PROBLEM SOLVING

For See Exercises Example

16–18 1 19–24 2 25 3 26–29 4

Independent Practice Find all possible values of each expression.

16. cos -1 1 17. sin -1 √ " 3 _ 2

18. tan -1 (-1)

Evaluate each inverse trigonometric function. Give your answer in both radians and degrees.

19. Sin -1 √ " 3 _ 2

20. Cos -1 (-1) 21. Tan -1 (- √ " 3 _ 3

)

22. Cos -1 (- √ " 3 _ 2

) 23. Tan -1 √ " 3 24. Sin -1 √ " 3

25. Volleyball A volleyball player spikes the ball from a height of 2.44 m. Assume that the path of the ball is a straight line. To the nearest degree, what is the maximum angle θ at which the ball can be hit and land within the court?

Solve each equation to the nearest tenth. Use the given restrictions.

26. sin θ = -0.75, for -90° ≤ θ ≤ 90° 27. sin θ = -0.75, for 180° < θ < 270°

28. cos θ = 0.1, for 0° ≤ θ ≤ 180° 29. cos θ = 0.1, for 270° < θ < 360°

30. Aviation The pilot of a small plane is flying at an altitude of 2000 ft. The pilot plans to start the final descent toward a runway when the horizontal distance between the plane and the runway is 2 mi. To the nearest degree, what will be the angle of depression θ from the plane to the runway at this point?

31. Multi-Step The table shows the

Pool Style

Length(ft)

Shallow End Depth

(ft)Deep End Depth (ft)

A 38 3 8

B 25 2 6

C 50 2.5 7

dimensions of three pool styles offered by a construction company.a. To the nearest tenth of a degree,

what angle θ does the bottom of each pool make with the horizontal?

b. Which pool style’s bottom has the steepest slope? Explain.

c. What if...? If the slope of the bottom of a pool can be no greater than 1 __ 6 , what is the greatest angle θ that the bottom of the pool can make with the horizontal? Round to the nearest tenth of a degree.

32. Navigation Lines of longitude are closer together near the poles than at the equator. The formula for the length " of 1° of longitude in miles is " = 69.0933 cos θ, where θ is the latitude in degrees.

a. At what latitude, to the nearest degree, is the length of a degree of longitude approximately 59.8 miles?

b. To the nearest mile, how much longer is the length of a degree of longitude at the equator, which has a latitude of 0°, than at the Arctic Circle, which has a latitude of about 66°N?

Skills Practice p. S29Application Practice p. S44

Extra Practice

a207se_c13l04_0950_0955.indd 954a207se_c13l04_0950_0955.indd 954 11/21/05 2:47:05 PM11/21/05 2:47:05 PM

           

954 Chapter 13 Trigonometric Functions

A flight simulator is a device used in training pilots that mimics flight conditions as realistically as possible. Some flight simulators involve full-size cockpits equipped with sound, visual, and motion systems.

Aviation

PRACTICE AND PROBLEM SOLVING

For See Exercises Example

16–18 1 19–24 2 25 3 26–29 4

Independent Practice Find all possible values of each expression.

16. cos -1 1 17. sin -1 √ " 3 _ 2

18. tan -1 (-1)

Evaluate each inverse trigonometric function. Give your answer in both radians and degrees.

19. Sin -1 √ " 3 _ 2

20. Cos -1 (-1) 21. Tan -1 (- √ " 3 _ 3

)

22. Cos -1 (- √ " 3 _ 2

) 23. Tan -1 √ " 3 24. Sin -1 √ " 3

25. Volleyball A volleyball player spikes the ball from a height of 2.44 m. Assume that the path of the ball is a straight line. To the nearest degree, what is the maximum angle θ at which the ball can be hit and land within the court?

Solve each equation to the nearest tenth. Use the given restrictions.

26. sin θ = -0.75, for -90° ≤ θ ≤ 90° 27. sin θ = -0.75, for 180° < θ < 270°

28. cos θ = 0.1, for 0° ≤ θ ≤ 180° 29. cos θ = 0.1, for 270° < θ < 360°

30. Aviation The pilot of a small plane is flying at an altitude of 2000 ft. The pilot plans to start the final descent toward a runway when the horizontal distance between the plane and the runway is 2 mi. To the nearest degree, what will be the angle of depression θ from the plane to the runway at this point?

31. Multi-Step The table shows the

Pool Style

Length(ft)

Shallow End Depth

(ft)Deep End Depth (ft)

A 38 3 8

B 25 2 6

C 50 2.5 7

dimensions of three pool styles offered by a construction company.a. To the nearest tenth of a degree,

what angle θ does the bottom of each pool make with the horizontal?

b. Which pool style’s bottom has the steepest slope? Explain.

c. What if...? If the slope of the bottom of a pool can be no greater than 1 __ 6 , what is the greatest angle θ that the bottom of the pool can make with the horizontal? Round to the nearest tenth of a degree.

32. Navigation Lines of longitude are closer together near the poles than at the equator. The formula for the length " of 1° of longitude in miles is " = 69.0933 cos θ, where θ is the latitude in degrees.

a. At what latitude, to the nearest degree, is the length of a degree of longitude approximately 59.8 miles?

b. To the nearest mile, how much longer is the length of a degree of longitude at the equator, which has a latitude of 0°, than at the Arctic Circle, which has a latitude of about 66°N?

Skills Practice p. S29Application Practice p. S44

Extra Practice

a207se_c13l04_0950_0955.indd 954a207se_c13l04_0950_0955.indd 954 11/21/05 2:47:05 PM11/21/05 2:47:05 PM

             

954 Chapter 13 Trigonometric Functions

A flight simulator is a device used in training pilots that mimics flight conditions as realistically as possible. Some flight simulators involve full-size cockpits equipped with sound, visual, and motion systems.

Aviation

PRACTICE AND PROBLEM SOLVING

For See Exercises Example

16–18 1 19–24 2 25 3 26–29 4

Independent Practice Find all possible values of each expression.

16. cos -1 1 17. sin -1 √ " 3 _ 2

18. tan -1 (-1)

Evaluate each inverse trigonometric function. Give your answer in both radians and degrees.

19. Sin -1 √ " 3 _ 2

20. Cos -1 (-1) 21. Tan -1 (- √ " 3 _ 3

)

22. Cos -1 (- √ " 3 _ 2

) 23. Tan -1 √ " 3 24. Sin -1 √ " 3

25. Volleyball A volleyball player spikes the ball from a height of 2.44 m. Assume that the path of the ball is a straight line. To the nearest degree, what is the maximum angle θ at which the ball can be hit and land within the court?

Solve each equation to the nearest tenth. Use the given restrictions.

26. sin θ = -0.75, for -90° ≤ θ ≤ 90° 27. sin θ = -0.75, for 180° < θ < 270°

28. cos θ = 0.1, for 0° ≤ θ ≤ 180° 29. cos θ = 0.1, for 270° < θ < 360°

30. Aviation The pilot of a small plane is flying at an altitude of 2000 ft. The pilot plans to start the final descent toward a runway when the horizontal distance between the plane and the runway is 2 mi. To the nearest degree, what will be the angle of depression θ from the plane to the runway at this point?

31. Multi-Step The table shows the

Pool Style

Length(ft)

Shallow End Depth

(ft)Deep End Depth (ft)

A 38 3 8

B 25 2 6

C 50 2.5 7

dimensions of three pool styles offered by a construction company.a. To the nearest tenth of a degree,

what angle θ does the bottom of each pool make with the horizontal?

b. Which pool style’s bottom has the steepest slope? Explain.

c. What if...? If the slope of the bottom of a pool can be no greater than 1 __ 6 , what is the greatest angle θ that the bottom of the pool can make with the horizontal? Round to the nearest tenth of a degree.

32. Navigation Lines of longitude are closer together near the poles than at the equator. The formula for the length " of 1° of longitude in miles is " = 69.0933 cos θ, where θ is the latitude in degrees.

a. At what latitude, to the nearest degree, is the length of a degree of longitude approximately 59.8 miles?

b. To the nearest mile, how much longer is the length of a degree of longitude at the equator, which has a latitude of 0°, than at the Arctic Circle, which has a latitude of about 66°N?

Skills Practice p. S29Application Practice p. S44

Extra Practice

a207se_c13l04_0950_0955.indd 954a207se_c13l04_0950_0955.indd 954 11/21/05 2:47:05 PM11/21/05 2:47:05 PM