4-3 Trigonometric Functions on the Unit...

The given point lies on the terminal side of an angle θ in standard position. Find the values of the six

trigonometric functions of θ. 3. (−4, −3)

SOLUTION: Use the values of x and y to find r.

Use x = , y = , and r = 5 to write the six trigonometric ratios.

6. (5, −3)

Use x = 5, y = , and r = to write the six trigonometric ratios.

Find the exact value of each trigonometric function, if defined. If not defined, write undefined.

9. sin

SOLUTION:

The terminal side of in standard position lies on the positive y-axis. Choose a point P(0, 1) on the terminal side of

the angle because r = 1.

12. csc 270°

SOLUTION:

The terminal side of in standard position lies on the negative y-axis. Choose a point P(0, ) on the terminal side of the angle because r = 1.

15. tan π

SOLUTION:

The terminal side of π in standard position lies on the negative x-axis. Choose a point P( , 0) on the terminal side of the angle because r = 1.

Sketch each angle. Then find its reference angle.18. 135°

SOLUTION:

The terminal side of 135º lies in Quadrant II. Therefore, its reference angle is θ ' = 180º – 135º or 45º.

21. −405°

SOLUTION:

A coterminal angle is −405° + 360(2)° or 315°. The terminal side of 315° lies in Quadrant IV, so its reference angle is 360º – 315º or 45º.

SOLUTION:

A coterminal angle is + 2(−1)π or The terminal side of lies in Quadrant I, so the reference angle is

Find the exact value of each expression.

27. sin

SOLUTION:

Because the terminal side of θ lies in Quadrant II, the reference angle θ ' is or .

In Quadrant II, sin θ is positive and .

30. sec (−150°)

SOLUTION:

A coterminal angle is −150° + 360° or 210°, which lies in Quadrant III. Because the terminal side of θ lies in Quadrant III. So, the reference angle θ ' is 210º – 180º or 30º.

Because secant and cosine are reciprocal functions and cos θ is negative in Quadrant III, it follows that sec θ is alsonegative in Quadrant III.

Find the exact values of the five remaining trigonometric functions of θ.

33. tan θ = 2, where sin θ > 0 and cos θ > 0

SOLUTION:

To find the other function values, you must find the coordinates of a point on the terminal side of θ. You know that

sin θ and cos θ are positive, so θ must lie in Quadrant I. This means that both x and y are positive.

Because tan θ = or , use the point (1, 2) to find r.

Use x = 1, y = 2, and r = to write the five remaining trigonometric ratios.

SOLUTION:

sin θ and cos θ are negative, so θ must lie in Quadrant III. This means that both x and y are negative.

Because cos θ = or , use the point ( , y) and r = 13 to find y .

Use x = , y = , and r = 13 to write the five remaining trigonometric ratios.

39. tan θ = −1, where sin θ < 0

SOLUTION:

sin θ is negative and cos θ is positive, so θ must lie in Quadrant IV. This means that x is positive and y is negative.

Because tan θ = or , use the point ( , ) to find r.

Use x = , y = , and r = to write the five remaining trigonometric ratios.

42. COIN FUNNEL A coin is dropped into a funnel where it spins in smaller circles until it drops into the bottom of the bank. The diameter of the first circle the coin makes is 24 centimeters. Before completing one full circle, the

coin travels 150° and falls over. What is the new position of the coin relative to the center of the funnel?

SOLUTION: Let the center of the funnel represent the origin on the coordinate plane and the final position of the coin have coordinates (x, y). The definitions of sine and cosine can then be used to find the values of x and y . The value of r, 12 cm, is the length of the radius of the first circle. The coin travels through an angle of 150º, so the reference angle is 180º – 150º or 30º. Since the final position of the coin corresponds to Quadrant II, the cosine of 150º is negative and the sine of 150º is positive.

Therefore, the coordinates of the final position of the coin are or about (–10.4, 6).

Find the exact value of each expression. If undefined, write undefined.

45. cos

SOLUTION:

Rewrite as the sum of and

48. cot 510°

SOLUTION: Rewrite 510° as the sum of 150° and 2 times 180°.

SOLUTION:

Rewrite as a sum of and .

54. sec

SOLUTION:

corresponds to the point (x, y) = on the unit circle.

57. tan

SOLUTION:

Rewrite as the sum of and 3 times .

Complete each trigonometric expression.

60. cos 60° = sin ___

SOLUTION:

60° corresponds to the point (x, y) = on the unit circle. So, cos 60° = .

On the unit circle, sin 30° = and sin 150° = . Therefore, cos 60° = sin 30° or cos 60° = sin 150° .

63. cos = sin ___

SOLUTION:

corresponds to the point (x, y) = on the unit circle. So, cos =

On the unit circle, sin = and sin = . Therefore, cos = sin or cos = sin .

66. ICE CREAM The monthly sales in thousands of dollars for Fiona’s Fine Ice Cream shop can be modeled by

, where t = 1 represents January, t = 2 represents February, and so on.

a. Estimate the sales for January, March, July, and October. b. Describe why the ice cream shop’s sales can be represented by a trigonometric function.

SOLUTION: a. January corresponds to t = 1.

March corresponds to t = 3.

July corresponds to t = 7.

October corresponds to t = 10.

b. Sample answer: The ice cream shop’s sales can be represented by a trigonometric function because people eat more ice cream in the summer and less in the winter.

Use the given values to evaluate the trigonometric functions.

69. sec θ = ; cos θ = ?; cos (−θ) = ?

SOLUTION:

Find the coordinates of P for each circle with the given radius and angle measure.

SOLUTION: Use the definitions of the cosine and sine functions to find the values of x and y . Because P is in Quadrant IV, the

cosine of is positive and the sine of is negative. The reference angle for is and the radius r is 3.

So, the coordinates of P are .

SOLUTION: Use the definitions of the cosine and sine functions to find the values of x and y . Because P is in Quadrant III, the

cosine and sine of are negative. The reference angle for is and the radius r is 8.

So, the coordinates of P are (−4, −4 ).

eSolutions Manual - Powered by Cognero Page 1

4-3 Trigonometric Functions on the Unit Circle

6. (5, −3)

9. sin

SOLUTION:

12. csc 270°

SOLUTION:

15. tan π

SOLUTION:

21. −405°

SOLUTION:

27. sin

SOLUTION:

30. sec (−150°)

SOLUTION:

45. cos

SOLUTION:

48. cot 510°

SOLUTION:

54. sec

SOLUTION:

57. tan

SOLUTION:

60. cos 60° = sin ___

SOLUTION:

63. cos = sin ___

SOLUTION:

69. sec θ = ; cos θ = ?; cos (−θ) = ?

SOLUTION:

6. (5, −3)

9. sin

SOLUTION:

12. csc 270°

SOLUTION:

15. tan π

SOLUTION:

21. −405°

SOLUTION:

27. sin

SOLUTION:

30. sec (−150°)

SOLUTION:

45. cos

SOLUTION:

48. cot 510°

SOLUTION:

54. sec

SOLUTION:

57. tan

SOLUTION:

60. cos 60° = sin ___

SOLUTION:

63. cos = sin ___

SOLUTION:

69. sec θ = ; cos θ = ?; cos (−θ) = ?

SOLUTION:

6. (5, −3)

9. sin

SOLUTION:

12. csc 270°

SOLUTION:

15. tan π

SOLUTION:

21. −405°

SOLUTION:

27. sin

SOLUTION:

30. sec (−150°)

SOLUTION:

45. cos

SOLUTION:

48. cot 510°

SOLUTION:

54. sec

SOLUTION:

57. tan

SOLUTION:

60. cos 60° = sin ___

SOLUTION:

63. cos = sin ___

SOLUTION:

69. sec θ = ; cos θ = ?; cos (−θ) = ?

SOLUTION:

6. (5, −3)

9. sin

SOLUTION:

12. csc 270°

SOLUTION:

15. tan π

SOLUTION:

21. −405°

SOLUTION:

27. sin

SOLUTION:

30. sec (−150°)

SOLUTION:

45. cos

SOLUTION:

48. cot 510°

SOLUTION:

54. sec

SOLUTION:

57. tan

SOLUTION:

60. cos 60° = sin ___

SOLUTION:

63. cos = sin ___

SOLUTION:

69. sec θ = ; cos θ = ?; cos (−θ) = ?

SOLUTION:

6. (5, −3)

9. sin

SOLUTION:

12. csc 270°

SOLUTION:

15. tan π

SOLUTION:

21. −405°

SOLUTION:

27. sin

SOLUTION:

30. sec (−150°)

SOLUTION:

45. cos

SOLUTION:

48. cot 510°

SOLUTION:

54. sec

SOLUTION:

57. tan

SOLUTION:

60. cos 60° = sin ___

SOLUTION:

63. cos = sin ___

SOLUTION:

69. sec θ = ; cos θ = ?; cos (−θ) = ?

SOLUTION:

6. (5, −3)

9. sin

SOLUTION:

12. csc 270°

SOLUTION:

15. tan π

SOLUTION:

21. −405°

SOLUTION:

27. sin

SOLUTION:

30. sec (−150°)

SOLUTION:

45. cos

SOLUTION:

48. cot 510°

SOLUTION:

54. sec

SOLUTION:

57. tan

SOLUTION:

60. cos 60° = sin ___

SOLUTION:

63. cos = sin ___

SOLUTION:

69. sec θ = ; cos θ = ?; cos (−θ) = ?

SOLUTION:

6. (5, −3)

9. sin

SOLUTION:

12. csc 270°

SOLUTION:

15. tan π

SOLUTION:

21. −405°

SOLUTION:

27. sin

SOLUTION:

30. sec (−150°)

SOLUTION:

45. cos

SOLUTION:

48. cot 510°

SOLUTION:

54. sec

SOLUTION:

57. tan

SOLUTION:

60. cos 60° = sin ___

SOLUTION:

63. cos = sin ___

SOLUTION:

69. sec θ = ; cos θ = ?; cos (−θ) = ?

SOLUTION:

6. (5, −3)

9. sin

SOLUTION:

12. csc 270°

SOLUTION:

15. tan π

SOLUTION:

21. −405°

SOLUTION:

27. sin

SOLUTION:

30. sec (−150°)

SOLUTION:

45. cos

SOLUTION:

48. cot 510°

SOLUTION:

54. sec

SOLUTION:

57. tan

SOLUTION:

60. cos 60° = sin ___

SOLUTION:

63. cos = sin ___

SOLUTION:

69. sec θ = ; cos θ = ?; cos (−θ) = ?

SOLUTION:

6. (5, −3)

9. sin

SOLUTION:

12. csc 270°

SOLUTION:

15. tan π

SOLUTION:

21. −405°

SOLUTION:

27. sin

SOLUTION:

30. sec (−150°)

SOLUTION:

45. cos

SOLUTION:

48. cot 510°

SOLUTION:

54. sec

SOLUTION:

57. tan

SOLUTION:

60. cos 60° = sin ___

SOLUTION:

63. cos = sin ___

SOLUTION:

69. sec θ = ; cos θ = ?; cos (−θ) = ?

SOLUTION:

6. (5, −3)

9. sin

SOLUTION:

12. csc 270°

SOLUTION:

15. tan π

SOLUTION:

21. −405°

SOLUTION:

27. sin

SOLUTION:

30. sec (−150°)

SOLUTION:

45. cos

SOLUTION:

48. cot 510°

SOLUTION:

54. sec

SOLUTION:

57. tan

SOLUTION:

60. cos 60° = sin ___

SOLUTION:

63. cos = sin ___

SOLUTION:

69. sec θ = ; cos θ = ?; cos (−θ) = ?

SOLUTION:

6. (5, −3)

9. sin

SOLUTION:

12. csc 270°

SOLUTION:

15. tan π

SOLUTION:

21. −405°

SOLUTION:

27. sin

SOLUTION:

30. sec (−150°)

SOLUTION:

45. cos

SOLUTION:

48. cot 510°

SOLUTION:

54. sec

SOLUTION:

57. tan

SOLUTION:

60. cos 60° = sin ___

SOLUTION:

63. cos = sin ___

SOLUTION:

69. sec θ = ; cos θ = ?; cos (−θ) = ?

SOLUTION:

6. (5, −3)

9. sin

SOLUTION:

12. csc 270°

SOLUTION:

15. tan π

SOLUTION:

21. −405°

SOLUTION:

27. sin

SOLUTION:

30. sec (−150°)

SOLUTION:

45. cos

SOLUTION:

48. cot 510°

SOLUTION:

54. sec

SOLUTION:

57. tan

SOLUTION:

60. cos 60° = sin ___

SOLUTION:

63. cos = sin ___

SOLUTION:

69. sec θ = ; cos θ = ?; cos (−θ) = ?

SOLUTION:

6. (5, −3)

9. sin

SOLUTION:

12. csc 270°

SOLUTION:

15. tan π

SOLUTION:

21. −405°

SOLUTION:

27. sin

SOLUTION:

30. sec (−150°)

SOLUTION:

45. cos

SOLUTION:

48. cot 510°

SOLUTION:

54. sec

SOLUTION:

57. tan

SOLUTION:

60. cos 60° = sin ___

SOLUTION:

63. cos = sin ___

SOLUTION:

69. sec θ = ; cos θ = ?; cos (−θ) = ?

SOLUTION:

4-3 Trigonometric Functions on the Unit...

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