5-1 Trigonometric Identities - Montville Township Public ...3/(5(35(6(17$7,216 In this problem, you...

Find the value of each expression using the given information.

1. If cot = , find tan .

SOLUTION:

2. If cos x = , find sec x.

SOLUTION:

3. If tan = , find cot .

SOLUTION:

4. If sin = , find csc .

SOLUTION:

5. If cos x = and sin x = , find cot x.

SOLUTION:

6. If sec = 2 and tan φ = , find sin .

SOLUTION:

Use the reciprocal identity to find

Then use the quotient identity to find

7. If csc = and cot = , find sec .

SOLUTION:

Use the reciprocal identity to find sin

Use the quotient identity to find cos α.

Use the reciprocal identity to find sec

8. If sec = 8 and tan = 3 , find csc .

SOLUTION:

Use the reciprocal identity to find cos

Use the quotient identity to find sin .

Use the reciprocal identity to find csc

9. sec and cos ; tan θ = –5, cos > 0

SOLUTION: Use the Pythagorean Identity that involves tan θ to find sec θ.

Since we are given that cos θ is positive, the reciprocal function sec θ must be positive, so

cos θ.

10. cot and sec ; sin θ = , tan < 0

SOLUTION:

Use the Pythagorean Identity that involves sin to find cos .

Since tan θ = is negative, and sin θ is positive,

then cos must be negative. Therefore,

Use the quotient identity to find cot .

11. tan and sin ; sec = 4, sin > 0

SOLUTION:

Use the Pythagorean Identity that involves sec to find tan .

Since sin θ is positive, and is positive, tan

θ must be positive. Therefore,

12. sin and cot ; cos = , sin < 0

SOLUTION:

Use the Pythagorean Identity that involves cos to find sin .

Since sin θ is negative, .

13. cos and tan ; csc = , tan > 0

SOLUTION:

Use the Pythagorean Identity that involves csc to find cot .

Since we are given that tan θ is positive, cot θ must

also be positive. Therefore, .

Use the reciprocal identity to find tan

Use the quotient identity to find cos .

14. sin and cos ; cot = 8, csc < 0

SOLUTION:

Use the Pythagorean Identity that involves cot to find csc .

Since we are given that csc θ is negative, csc θ =

15. cot and sin ; sec = , sin > 0

SOLUTION: Use the Pythagorean Identity that involves sec θ to

find tan .

Since we are given that sec is negative, we know that cos is negative. We are also given that sin θ is positive. Since cos is negative and sin is positive, tan is negative. Therefore,

Use the reciprocal identity to find cot

Then use the Pythagorean Identity that involves cos θ to find sin θ.

Since we are given that sin θ is positive, sin θ =

16. tan and csc ; cos = , sin < 0

SOLUTION: Use the Pythagorean Identity that involves cos θ to find sin θ.

Since we are given that sin θ is negative, we know

that sin θ = . Use the reciprocal identity

to find csc θ.

Use the quotient identity to find tan θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

Simplify each expression.22. csc x sec x – tan x

SOLUTION:

23. csc x – cos x cot x

SOLUTION:

24. sec x cot x – sin x

SOLUTION:

30. cot x – csc2 x cot x

SOLUTION:

31. cot x – cos3 x csc x

SOLUTION:

Simplify each expression.

SOLUTION:

37. SUNGLASSES Many sunglasses are made with polarized lenses, which reduce the intensity of light. The intensity of light emerging from a system of two

polarizing lenses I can be calculated by I = I0 –

, where I0 is the intensity of light entering the

system of lenses and θ is the angle of the axis of the second lens in relation to that of the first lens.

a. Simplify the formula for the intensity of light emerging from a system of two polarized lenses. b. If a pair of sunglasses contains a system of two polarizing lenses with axes at 30º to one another, what proportion of the intensity of light entering the sunglasses emerges?

SOLUTION: a.

b. Evaluate I = I0 cos

2 θ for θ = 30°.

Therefore, of the intensity of light entering the

sunglasses emerges.

Rewrite as an expression that does not involve a fraction.

SOLUTION:

Determine whether each parent trigonometric function shown is odd or even. Explain your reasoning.

SOLUTION: Functions that are symmetric with respect to the y-axis are even function. Functions that are symmetric with respect to the origin are odd functions. The graph shown is an odd function. The graph is f (x) = cot x. f (–x) = cot (–x) = –cot x =–f (x). Since f (–x) = –f (x), f (x) = cot x is an odd function.

SOLUTION: Functions that are symmetric with respect to the y-axis are even function. The graph shown is an even function. The graph is f (x) = sec x. f (–x) = sec (–x) = sec x =f(x). Since f (–x) = f (x), f (x) = sec x is an even function.

50. SOCCER When a soccer ball is kicked from the ground, its height y and horizontal displacement x are

related by , where v0 is

the initial velocity of the ball, is the angle at which it was kicked, and g is the acceleration due to

gravity. Rewrite this equation so that tan is the only trigonometric function that appears in the equation.

SOLUTION:

Write each expression in terms of a single trigonometric function.

51. tan x – csc x sec x

SOLUTION:

52. cos x + tan x sin x

SOLUTION:

53. csc x tan2 x – sec

2 x csc x

SOLUTION:

54. sec x csc x – cos x csc x

SOLUTION:

55. MULTIPLE REPRESENTATIONS In this problem, you will investigate the verification of trigonometric identities. Consider the functions shown. i.

a. TABULAR Copy and complete the table below,without graphing the functions.

b. GRAPHICAL Graph each function on a graphing calculator. c. VERBAL Make a conjecture about the relationship between y1 and y2. Repeat for y3 and

d. ANALYTICAL Are the conjectures that you made in part c valid for the entire domain of each function? Explain your reasoning.

SOLUTION: a. Evaluate for , , 0, , and

Evaluate for

Evaluate for , , 0, ,

Evaluate for , , 0, , and

Complete the table.

b. To graph y1, enter the following into Y1 and then

press Zoom Trig.

To graph y2, enter the following into Y1 and then

press Zoom Trig.

c. The graphs for y1 and y2 are not the same, so y1

y2. For the window shown, the graphs of y3 and

y4 appear identical. Therefore it appears that y3 =

d. yes

Since y2 = , and for

the entire domain of each function, y1 y2 .

Since y4 = = y3, y3 = y4 for the

entire domain of each function.

Rewrite each expression as a single logarithm and simplify the answer.

56. ln |sin x| – ln |cos x|

SOLUTION:

57. ln |sec x| – ln |cos x|

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

59. ln (sec2 x – tan

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

60. ELECTRICITY A current in a wire in a magnetic field causes a force to act on the wire. The strength of the magnetic field can be determined using the

formula , where F is the force on the

wire, I is the current in the wire, is the length of

the wire, and θ is the angle the wire makes with the magnetic field. Some physics books give the formula

as F = I Bsin θ. Show that the two formulas are equivalent.

SOLUTION: Start with one equation and use algebraic and trigonometric identities to form the other.

61. LIGHT WAVES When light shines through two narrow slits, a series of light and dark fringes appear.The angle θ, in radians, locating the mth fringe can

be calculated by sin θ = , where d is the

distance between the two slits, and λ is the wavelength of light.

a. Rewrite the formula in terms of csc θ. b. Determine the angle locating the 100th fringe when light having a wavelength of 550 nanometers isshined through double slits spaced 0.5 millimeters apart.

SOLUTION: a.

b. Solve the equation when m = 100,

= 550 nanometers, and d = 0.5 mm. To do this, first convert 0.5 mm to nanometers. One nanometer

is 1 × 10–9

62. PROOF Prove that the area of the triangle is A =

s = (a + b + c).

(Hint: The area of an oblique triangle is A = bc

sin A.)

SOLUTION:

By the Law of Cosines, , so

Rewrite each factor under the radical using this value for cos A.

Rewrite each of these expression in terms of s if s =

Substitute these value into the equation

63. ERROR ANALYSIS Jenelle and Chloe are

simplifying . Jenelle thinks that the

expression simplifies to , and Chloe

thinks that it simplifies to csc2 x – tan

2 x. Is either of

them correct? Explain your reasoning.

SOLUTION: Jenelle is correct.

Chloe incorrectly simplified as

CHALLENGE Write each of the basic trigonometric functions in terms of the following functions.

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

eSolutions Manual - Powered by Cognero Page 1

5-1 Trigonometric Identities

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

SOLUTION:

cos θ.

SOLUTION:

find tan .

to find csc θ.

17. If csc = –1.24, find .

SOLUTION:

18. If cos x = 0.61, find .

SOLUTION:

19. If tan = –1.52, find .

SOLUTION:

20. If sin = 0.18, find .

SOLUTION:

21. If cot x = 1.35, find .

SOLUTION:

SOLUTION: a.

2 θ for θ = 30°.

sunglasses emerges.

SOLUTION:

2 x csc x

SOLUTION:

Evaluate for

Complete the table.

press Zoom Trig.

d. yes

SOLUTION:

58. ln (cot2 x + 1) + ln |sec x|

SOLUTION:

2 x) – ln (1 – cos

SOLUTION:

Alternate solution:

SOLUTION: a.

is 1 × 10–9

s = (a + b + c).

sin A.)

SOLUTION:

2 x. Is either of

64. sin x

SOLUTION:

csc x =

cos x:

sec x:

tan x:

cot x:

65. cos x

SOLUTION: sin x:

csc x:

sec x: sec x =

tan x:

cot x:

66. tan x

SOLUTION: cot x =

sec x:

cos x:

csc x:

sin x:

REASONING Determine whether each statement is true or false . Explain your reasoning.

67. csc2 x tan x = csc x sec x is true for all real

numbers.

SOLUTION:

False; The expressions csc2 x tan x and csc x sec x

are equivalent for all values of x for which both expressions are defined, as shown below.

However, these expressions are not defined for all

real numbers. For example, when x = , csc2 x tan

x and csc x sec x are undefined.

68. The odd-even identities can be used to prove that thegraphs of y = cos x and y = sec x are symmetric with respect to the y-axis.

SOLUTION: True; sample answer: The graphs of even functions are symmetric with respect to the y-axis. Since cos xand sec x are both even functions, the graphs of these functions are therefore symmetric with respectto the y-axis.

PROOF Prove each Pythagorean identity.

69. tan2 + 1 = sec

SOLUTION:

According to the Pythagorean Theorem, x2 + y

r2. When each side of the equation is divided by x

1 + = . Since tan = and sec = , 1 +

= is equivalent to 1 + tan2 = sec

tan2 + 1 = sec2

70. cot2 + 1 = csc

SOLUTION:

When each side of the equation is divided by y2,

+ 1 = . Since cot = and csc = ,,

+ 1 = is equivalent to cot2 + 1 = csc

or 1 + cot2 = csc2 .

71. PREWRITE Use a chart or a table to help you organize the major trigonometric identities found in Lesson 5-1.

SOLUTION: Reciprocal Identities sin θ =

cos θ =

tan θ =

csc θ =

sec θ =

cot θ =

Quotient Identities

tan θ =

cot θ =

Pythagorean Identities sin

2 θ + cos

2 θ = 1

tan2 θ + 1 = sec2 θ

cot2 θ + 1 = csc

Cofunction Identities

Odd-Even Identities

sin (–θ) = –sin θ csc (–θ) = –csc θ cos (–θ) = cos θ sec (–θ) = sec θ tan (–θ) = –tan θ cot (–θ) = –cot θ

Solve each triangle. Round to the nearest tenth, if necessary.

SOLUTION:

Because two angles are given, B = 180 – (62 +

17 ) or 101 . Use the Law of Sines to find b and c.

Therefore, B = 101 , c 3.0, and b 3.4.

SOLUTION:

Because two angles are given, C = 180 – (59 +4

or 73 . Use the Law of Sines to find a and b.

Therefore, C = 73 , a 55.6, and b 48.2.

SOLUTION: Use the Law of Sines to find b,and C.

Find B, B = 180º – (122º +37º) or 21º.

Therefore, B 21 , C 37 , and b 13.1.

SOLUTION: Use the Law of Sines to find c,and B.

Find C, C = 180º – (65º +46º) or 69º.

Therefore, B 46 , C 69 , and c 5.2.

SOLUTION: Because two angles are given, C = 180º – (63º + 20º97º. Use the Law of Sines to find a and b.

SOLUTION: Use the Law of Sines to find a,and A.

Find B,B = 180º – (40º +75º) or 65º.

Therefore, A 40 , B 65 , and b 2.8.

Find the exact value of each expression, if it exists.

78. cot

SOLUTION:

Let u = , so u = .

Because the domain of the inverse sine function is restricted to Quadrants I and IV, u must line in one of these quadrants. The solution is similar for each quadrant, so we will solve for Quadrant I. Draw a diagram of a right triangle with an acute angle u, an opposite side length 7 and a hypotenuse length 9.

The length of the side adjacent to u is

or . Now, solve for cot u.

So, cot = .

79. tan (arctan 3)

SOLUTION: The inverse property applies, because 3 lies on the

interval . Therefore, = 3.

80. cos

SOLUTION:

The inverse property applies, because lies on

the interval [–1, 1]. Therefore, =

SOLUTION:

Let u = cos– 1

and substitute.

Now find . Draw a diagram of a right

triangle with an acute angle u, an adjacent side

length and a hypotenuse of length 2.

The length of the side opposite u is

. Using this triangle,

we find that sin u = . Therefore,

82. cos – 1

SOLUTION:

Let u = . Then u = .

Because the domain of the inverse sine function is restricted to Quadrants I and IV, u must line in one of these quadrants. The solution is similar for each quadrant, so we will solve for Quadrant I. Draw a diagram of a right triangle with an acute

angle u, an opposite side length and a hypotenuse of length 2.

. Since is not a real

number, u does not exist. Therefore, does not

exist, so cos – 1

does not exist.

83. sin

SOLUTION:

Because the domain of the inverse cosine function isrestricted to Quadrants I and II, u must line in one ofthese quadrants. The solution is similar for each quadrant, so we will solve for Quadrant I. Draw a diagram of a right triangle with an acute angle u, an adjacent side length 3 and a hypotenuse length 5.

The length of the side opposite to u is

So, sin = sin u = .

84. ANTHROPOLOGY Allometry is the study of the relationship between the size of an organism and the size of any of its parts. A researcher decided to test for an allometry between the size of the human headcompared to the human body as a person ages. The data in the table represent the average American male. a. Find a quadratic model relating these data by linearizing the data and finding the linear regression equation. b. Use the model for the linearized data to find a model for the original data. c. Use your model to predict the height an Americanmale whose head circumference is 24 inches.

SOLUTION: a. For a quadratic model, replace (x, y) with (x, ln y).

Find the regression equation of the linearized data.

The rounded equation is .

Graph the linearized data.

b. Replace with ln y and simplify.

c. Substitute 24 for x.

The height is approximately 59.7 inches.

x y ln y 14.1 19.5 2.97 18.0 26.4 3.27 18.3 29.7 3.39 18.7 32.3 3.48 19.1 34.4 3.54 19.4 36.2 3.59 19.6 37.7 3.63

Let U = 0, 1, 2, 3, 4, 5, A = 6, 9, B = 6, 9, 10, C = 0, 1, 6, 9, 11, D = 2, 5, 11. Determine whether each statement is true or false . Explain your reasoning.

85. A ⊂ B

SOLUTION: True; all of the elements of A are also elements of B.

86. D ⊂ U

SOLUTION: False; all of the elements of D are not elements of U; 11 is an element of D but not an element of U.

87. SAT/ACT If x > 0 then

A (x + 1)

B (x – 1)2

C 3x – 1 D 3x

E 3(x – 1)2

SOLUTION:

88. REVIEW If sin x = m and 0 < x < 90°, then tan x =

SOLUTION:

89. Which of the following is equivalent to

⋅ tan θ ?

A tan θ B cot θ C sin θ D cos θ

SOLUTION:

The correct answer is choice B.

90. REVIEW Refer to the figure. If cos D = 0.8, what

is the length of ?

F 5 G 4 H 3.2

SOLUTION:

tan x:

cot x:

66. tan x

SOLUTION: cot x =

sec x:

cos x:

csc x:

sin x:

numbers.

SOLUTION:

69. tan2 + 1 = sec

SOLUTION:

tan2 + 1 = sec2

70. cot2 + 1 = csc

SOLUTION:

cos θ =

tan θ =

csc θ =

sec θ =

cot θ =

Quotient Identities

tan θ =

cot θ =

2 θ + cos

2 θ = 1

cot2 θ + 1 = csc

Odd-Even Identities

SOLUTION:

Find B, B = 180º – (122º +37º) or 21º.

Find C, C = 180º – (65º +46º) or 69º.

Find B,B = 180º – (40º +75º) or 65º.

78. cot

SOLUTION:

Let u = , so u = .

So, cot = .

79. tan (arctan 3)

80. cos

SOLUTION:

Let u = cos– 1

and substitute.

82. cos – 1

SOLUTION:

exist, so cos – 1

does not exist.

83. sin

SOLUTION:

So, sin = sin u = .

x y ln y 14.1 19.5 2.97 18.0 26.4 3.27 18.3 29.7 3.39 18.7 32.3 3.48 19.1 34.4 3.54 19.4 36.2 3.59 19.6 37.7 3.63

85. A ⊂ B

86. D ⊂ U

A (x + 1)

B (x – 1)2

C 3x – 1 D 3x

E 3(x – 1)2

SOLUTION:

⋅ tan θ ?

SOLUTION:

is the length of ?

F 5 G 4 H 3.2

SOLUTION:

tan x:

cot x:

66. tan x

SOLUTION: cot x =

sec x:

cos x:

csc x:

sin x:

numbers.

SOLUTION:

69. tan2 + 1 = sec

SOLUTION:

tan2 + 1 = sec2

70. cot2 + 1 = csc

SOLUTION:

cos θ =

tan θ =

csc θ =

sec θ =

cot θ =

Quotient Identities

tan θ =

cot θ =

2 θ + cos

2 θ = 1

cot2 θ + 1 = csc

Odd-Even Identities

SOLUTION:

Find B, B = 180º – (122º +37º) or 21º.

Find C, C = 180º – (65º +46º) or 69º.

Find B,B = 180º – (40º +75º) or 65º.

78. cot

SOLUTION:

Let u = , so u = .

So, cot = .

79. tan (arctan 3)

80. cos

SOLUTION:

Let u = cos– 1

and substitute.

82. cos – 1

SOLUTION:

exist, so cos – 1

does not exist.

83. sin

SOLUTION:

So, sin = sin u = .

x y ln y 14.1 19.5 2.97 18.0 26.4 3.27 18.3 29.7 3.39 18.7 32.3 3.48 19.1 34.4 3.54 19.4 36.2 3.59 19.6 37.7 3.63

85. A ⊂ B

86. D ⊂ U

A (x + 1)

B (x – 1)2

C 3x – 1 D 3x

E 3(x – 1)2

SOLUTION:

⋅ tan θ ?

SOLUTION:

is the length of ?

F 5 G 4 H 3.2

SOLUTION:

tan x:

cot x:

66. tan x

SOLUTION: cot x =

sec x:

cos x:

csc x:

sin x:

numbers.

SOLUTION:

69. tan2 + 1 = sec

SOLUTION:

tan2 + 1 = sec2

70. cot2 + 1 = csc

SOLUTION:

cos θ =

tan θ =

csc θ =

sec θ =

cot θ =

Quotient Identities

tan θ =

cot θ =

2 θ + cos

2 θ = 1

cot2 θ + 1 = csc

Odd-Even Identities

SOLUTION:

Find B, B = 180º – (122º +37º) or 21º.

Find C, C = 180º – (65º +46º) or 69º.

Find B,B = 180º – (40º +75º) or 65º.

78. cot

SOLUTION:

Let u = , so u = .

So, cot = .

79. tan (arctan 3)

80. cos

SOLUTION:

Let u = cos– 1

and substitute.

82. cos – 1

SOLUTION:

exist, so cos – 1

does not exist.

83. sin

SOLUTION:

So, sin = sin u = .

x y ln y 14.1 19.5 2.97 18.0 26.4 3.27 18.3 29.7 3.39 18.7 32.3 3.48 19.1 34.4 3.54 19.4 36.2 3.59 19.6 37.7 3.63

85. A ⊂ B

86. D ⊂ U

A (x + 1)

B (x – 1)2

C 3x – 1 D 3x

E 3(x – 1)2

SOLUTION:

⋅ tan θ ?

SOLUTION:

is the length of ?

F 5 G 4 H 3.2

SOLUTION:

tan x:

cot x:

66. tan x

SOLUTION: cot x =

sec x:

cos x:

csc x:

sin x:

numbers.

SOLUTION:

69. tan2 + 1 = sec

SOLUTION:

tan2 + 1 = sec2

70. cot2 + 1 = csc

SOLUTION:

cos θ =

tan θ =

csc θ =

sec θ =

cot θ =

Quotient Identities

tan θ =

cot θ =

2 θ + cos

2 θ = 1

cot2 θ + 1 = csc

Odd-Even Identities

SOLUTION:

Find B, B = 180º – (122º +37º) or 21º.

Find C, C = 180º – (65º +46º) or 69º.

Find B,B = 180º – (40º +75º) or 65º.

78. cot

SOLUTION:

Let u = , so u = .

So, cot = .

79. tan (arctan 3)

80. cos

SOLUTION:

Let u = cos– 1

and substitute.

82. cos – 1

SOLUTION:

exist, so cos – 1

does not exist.

83. sin

SOLUTION:

So, sin = sin u = .

x y ln y 14.1 19.5 2.97 18.0 26.4 3.27 18.3 29.7 3.39 18.7 32.3 3.48 19.1 34.4 3.54 19.4 36.2 3.59 19.6 37.7 3.63

85. A ⊂ B

86. D ⊂ U

A (x + 1)

B (x – 1)2

C 3x – 1 D 3x

E 3(x – 1)2

SOLUTION:

⋅ tan θ ?

SOLUTION:

is the length of ?

F 5 G 4 H 3.2

SOLUTION:

tan x:

cot x:

66. tan x

SOLUTION: cot x =

sec x:

cos x:

csc x:

sin x:

numbers.

SOLUTION:

69. tan2 + 1 = sec

SOLUTION:

tan2 + 1 = sec2

70. cot2 + 1 = csc

SOLUTION:

cos θ =

tan θ =

csc θ =

sec θ =

cot θ =

Quotient Identities

tan θ =

cot θ =

2 θ + cos

2 θ = 1

cot2 θ + 1 = csc

Odd-Even Identities

SOLUTION:

Find B, B = 180º – (122º +37º) or 21º.

Find C, C = 180º – (65º +46º) or 69º.

Find B,B = 180º – (40º +75º) or 65º.

78. cot

SOLUTION:

Let u = , so u = .

So, cot = .

79. tan (arctan 3)

80. cos

SOLUTION:

Let u = cos– 1

and substitute.

82. cos – 1

SOLUTION:

exist, so cos – 1

does not exist.

83. sin

SOLUTION:

So, sin = sin u = .

x y ln y 14.1 19.5 2.97 18.0 26.4 3.27 18.3 29.7 3.39 18.7 32.3 3.48 19.1 34.4 3.54 19.4 36.2 3.59 19.6 37.7 3.63

85. A ⊂ B

86. D ⊂ U

A (x + 1)

B (x – 1)2

C 3x – 1 D 3x

E 3(x – 1)2

SOLUTION:

⋅ tan θ ?

SOLUTION:

is the length of ?

F 5 G 4 H 3.2

SOLUTION:

tan x:

cot x:

66. tan x

SOLUTION: cot x =

sec x:

cos x:

csc x:

sin x:

numbers.

SOLUTION:

69. tan2 + 1 = sec

SOLUTION:

tan2 + 1 = sec2

70. cot2 + 1 = csc

SOLUTION:

cos θ =

tan θ =

csc θ =

sec θ =

cot θ =

Quotient Identities

tan θ =

cot θ =

2 θ + cos

2 θ = 1

cot2 θ + 1 = csc

Odd-Even Identities

SOLUTION:

Find B, B = 180º – (122º +37º) or 21º.

Find C, C = 180º – (65º +46º) or 69º.

Find B,B = 180º – (40º +75º) or 65º.

78. cot

SOLUTION:

Let u = , so u = .

So, cot = .

79. tan (arctan 3)

80. cos

SOLUTION:

Let u = cos– 1

and substitute.

82. cos – 1

SOLUTION:

exist, so cos – 1

does not exist.

83. sin

SOLUTION:

So, sin = sin u = .

x y ln y 14.1 19.5 2.97 18.0 26.4 3.27 18.3 29.7 3.39 18.7 32.3 3.48 19.1 34.4 3.54 19.4 36.2 3.59 19.6 37.7 3.63

85. A ⊂ B

86. D ⊂ U

A (x + 1)

B (x – 1)2

C 3x – 1 D 3x

E 3(x – 1)2

SOLUTION:

⋅ tan θ ?

SOLUTION:

is the length of ?

F 5 G 4 H 3.2

SOLUTION:

tan x:

cot x:

66. tan x

SOLUTION: cot x =

sec x:

cos x:

csc x:

sin x:

numbers.

SOLUTION:

69. tan2 + 1 = sec

SOLUTION:

tan2 + 1 = sec2

70. cot2 + 1 = csc

SOLUTION:

cos θ =

tan θ =

csc θ =

sec θ =

cot θ =

Quotient Identities

tan θ =

cot θ =

2 θ + cos

2 θ = 1

cot2 θ + 1 = csc

Odd-Even Identities

SOLUTION:

Find B, B = 180º – (122º +37º) or 21º.

Find C, C = 180º – (65º +46º) or 69º.

Find B,B = 180º – (40º +75º) or 65º.

78. cot

SOLUTION:

Let u = , so u = .

So, cot = .

79. tan (arctan 3)

80. cos

SOLUTION:

Let u = cos– 1

and substitute.

82. cos – 1

SOLUTION:

exist, so cos – 1

does not exist.

83. sin

SOLUTION:

So, sin = sin u = .

x y ln y 14.1 19.5 2.97 18.0 26.4 3.27 18.3 29.7 3.39 18.7 32.3 3.48 19.1 34.4 3.54 19.4 36.2 3.59 19.6 37.7 3.63

85. A ⊂ B

86. D ⊂ U

A (x + 1)

B (x – 1)2

C 3x – 1 D 3x

E 3(x – 1)2

SOLUTION:

⋅ tan θ ?

SOLUTION:

is the length of ?

F 5 G 4 H 3.2

SOLUTION:

tan x:

cot x:

66. tan x

SOLUTION: cot x =

sec x:

cos x:

csc x:

sin x:

numbers.

SOLUTION:

69. tan2 + 1 = sec

SOLUTION:

tan2 + 1 = sec2

70. cot2 + 1 = csc

SOLUTION:

cos θ =

tan θ =

csc θ =

sec θ =

cot θ =

Quotient Identities

tan θ =

cot θ =

2 θ + cos

2 θ = 1

cot2 θ + 1 = csc

Odd-Even Identities

SOLUTION:

Find B, B = 180º – (122º +37º) or 21º.

Find C, C = 180º – (65º +46º) or 69º.

Find B,B = 180º – (40º +75º) or 65º.

78. cot

SOLUTION:

Let u = , so u = .

So, cot = .

79. tan (arctan 3)

80. cos

SOLUTION:

Let u = cos– 1

and substitute.

82. cos – 1

SOLUTION:

exist, so cos – 1

does not exist.

83. sin

SOLUTION:

So, sin = sin u = .

x y ln y 14.1 19.5 2.97 18.0 26.4 3.27 18.3 29.7 3.39 18.7 32.3 3.48 19.1 34.4 3.54 19.4 36.2 3.59 19.6 37.7 3.63

85. A ⊂ B

86. D ⊂ U

A (x + 1)

B (x – 1)2

C 3x – 1 D 3x

E 3(x – 1)2

SOLUTION:

⋅ tan θ ?

SOLUTION:

is the length of ?

F 5 G 4 H 3.2

SOLUTION:

tan x:

cot x:

66. tan x

SOLUTION: cot x =

sec x:

cos x:

csc x:

sin x:

numbers.

SOLUTION:

69. tan2 + 1 = sec

SOLUTION:

tan2 + 1 = sec2

70. cot2 + 1 = csc

SOLUTION:

cos θ =

tan θ =

csc θ =

sec θ =

cot θ =

Quotient Identities

tan θ =

cot θ =

2 θ + cos

2 θ = 1

cot2 θ + 1 = csc

Odd-Even Identities

SOLUTION:

Find B, B = 180º – (122º +37º) or 21º.

Find C, C = 180º – (65º +46º) or 69º.

Find B,B = 180º – (40º +75º) or 65º.

78. cot

SOLUTION:

Let u = , so u = .

So, cot = .

79. tan (arctan 3)

80. cos

SOLUTION:

Let u = cos– 1

and substitute.

82. cos – 1

SOLUTION:

exist, so cos – 1

does not exist.

83. sin

SOLUTION:

So, sin = sin u = .

x y ln y 14.1 19.5 2.97 18.0 26.4 3.27 18.3 29.7 3.39 18.7 32.3 3.48 19.1 34.4 3.54 19.4 36.2 3.59 19.6 37.7 3.63

85. A ⊂ B

86. D ⊂ U

A (x + 1)

B (x – 1)2

C 3x – 1 D 3x

E 3(x – 1)2

SOLUTION:

⋅ tan θ ?

SOLUTION:

is the length of ?

F 5 G 4 H 3.2

SOLUTION:

tan x:

cot x:

66. tan x

SOLUTION: cot x =

sec x:

cos x:

csc x:

sin x:

numbers.

SOLUTION:

69. tan2 + 1 = sec

SOLUTION:

tan2 + 1 = sec2

70. cot2 + 1 = csc

SOLUTION:

cos θ =

tan θ =

csc θ =

sec θ =

cot θ =

Quotient Identities

tan θ =

cot θ =

2 θ + cos

2 θ = 1

cot2 θ + 1 = csc

Odd-Even Identities

SOLUTION:

Find B, B = 180º – (122º +37º) or 21º.

Find C, C = 180º – (65º +46º) or 69º.

Find B,B = 180º – (40º +75º) or 65º.

78. cot

SOLUTION:

Let u = , so u = .

So, cot = .

79. tan (arctan 3)

80. cos

SOLUTION:

Let u = cos– 1

and substitute.

82. cos – 1

SOLUTION:

exist, so cos – 1

does not exist.

83. sin

SOLUTION:

So, sin = sin u = .

x y ln y 14.1 19.5 2.97 18.0 26.4 3.27 18.3 29.7 3.39 18.7 32.3 3.48 19.1 34.4 3.54 19.4 36.2 3.59 19.6 37.7 3.63

85. A ⊂ B

86. D ⊂ U

A (x + 1)

B (x – 1)2

C 3x – 1 D 3x

E 3(x – 1)2

SOLUTION:

⋅ tan θ ?

SOLUTION:

is the length of ?

F 5 G 4 H 3.2

SOLUTION:

tan x:

cot x:

66. tan x

SOLUTION: cot x =

sec x:

cos x:

csc x:

sin x:

numbers.

SOLUTION:

69. tan2 + 1 = sec

SOLUTION:

tan2 + 1 = sec2

70. cot2 + 1 = csc

SOLUTION:

cos θ =

tan θ =

csc θ =

sec θ =

cot θ =

Quotient Identities

tan θ =

cot θ =

2 θ + cos

2 θ = 1

cot2 θ + 1 = csc

Odd-Even Identities

SOLUTION:

Find B, B = 180º – (122º +37º) or 21º.

Find C, C = 180º – (65º +46º) or 69º.

Find B,B = 180º – (40º +75º) or 65º.

78. cot

SOLUTION:

Let u = , so u = .

So, cot = .

79. tan (arctan 3)

80. cos

SOLUTION:

Let u = cos– 1

and substitute.

82. cos – 1

SOLUTION:

exist, so cos – 1

does not exist.

83. sin

SOLUTION:

So, sin = sin u = .

x y ln y 14.1 19.5 2.97 18.0 26.4 3.27 18.3 29.7 3.39 18.7 32.3 3.48 19.1 34.4 3.54 19.4 36.2 3.59 19.6 37.7 3.63

85. A ⊂ B

86. D ⊂ U

A (x + 1)

B (x – 1)2

C 3x – 1 D 3x

E 3(x – 1)2

SOLUTION:

⋅ tan θ ?

SOLUTION:

is the length of ?

F 5 G 4 H 3.2

SOLUTION:

tan x:

cot x:

66. tan x

SOLUTION: cot x =

sec x:

cos x:

csc x:

sin x:

numbers.

SOLUTION:

69. tan2 + 1 = sec

SOLUTION:

tan2 + 1 = sec2

70. cot2 + 1 = csc

SOLUTION:

cos θ =

tan θ =

csc θ =

sec θ =

cot θ =

Quotient Identities

tan θ =

cot θ =

2 θ + cos

2 θ = 1

cot2 θ + 1 = csc

Odd-Even Identities

SOLUTION:

Find B, B = 180º – (122º +37º) or 21º.

Find C, C = 180º – (65º +46º) or 69º.

Find B,B = 180º – (40º +75º) or 65º.

78. cot

SOLUTION:

Let u = , so u = .

So, cot = .

79. tan (arctan 3)

80. cos

SOLUTION:

Let u = cos– 1

and substitute.

82. cos – 1

SOLUTION:

exist, so cos – 1

does not exist.

83. sin

SOLUTION:

So, sin = sin u = .

x y ln y 14.1 19.5 2.97 18.0 26.4 3.27 18.3 29.7 3.39 18.7 32.3 3.48 19.1 34.4 3.54 19.4 36.2 3.59 19.6 37.7 3.63

85. A ⊂ B

86. D ⊂ U

A (x + 1)

B (x – 1)2

C 3x – 1 D 3x

E 3(x – 1)2

SOLUTION:

⋅ tan θ ?

SOLUTION:

is the length of ?

F 5 G 4 H 3.2

SOLUTION:

tan x:

cot x:

66. tan x

SOLUTION: cot x =

sec x:

cos x:

csc x:

sin x:

numbers.

SOLUTION:

69. tan2 + 1 = sec

SOLUTION:

tan2 + 1 = sec2

70. cot2 + 1 = csc

SOLUTION:

cos θ =

tan θ =

csc θ =

sec θ =

cot θ =

Quotient Identities

tan θ =

cot θ =

2 θ + cos

2 θ = 1

cot2 θ + 1 = csc

Odd-Even Identities

SOLUTION:

Find B, B = 180º – (122º +37º) or 21º.

Find C, C = 180º – (65º +46º) or 69º.

Find B,B = 180º – (40º +75º) or 65º.

78. cot

SOLUTION:

Let u = , so u = .

So, cot = .

79. tan (arctan 3)

80. cos

SOLUTION:

Let u = cos– 1

and substitute.

82. cos – 1

SOLUTION:

exist, so cos – 1

does not exist.

83. sin

SOLUTION:

So, sin = sin u = .

x y ln y 14.1 19.5 2.97 18.0 26.4 3.27 18.3 29.7 3.39 18.7 32.3 3.48 19.1 34.4 3.54 19.4 36.2 3.59 19.6 37.7 3.63

85. A ⊂ B

86. D ⊂ U

A (x + 1)

B (x – 1)2

C 3x – 1 D 3x

E 3(x – 1)2

SOLUTION:

⋅ tan θ ?

SOLUTION:

is the length of ?

F 5 G 4 H 3.2

SOLUTION:

tan x:

cot x:

66. tan x

SOLUTION: cot x =

sec x:

cos x:

csc x:

sin x:

numbers.

SOLUTION:

69. tan2 + 1 = sec

SOLUTION:

tan2 + 1 = sec2

70. cot2 + 1 = csc

SOLUTION:

cos θ =

tan θ =

csc θ =

sec θ =

cot θ =

Quotient Identities

tan θ =

cot θ =

2 θ + cos

2 θ = 1

cot2 θ + 1 = csc

Odd-Even Identities

SOLUTION:

Find B, B = 180º – (122º +37º) or 21º.

Find C, C = 180º – (65º +46º) or 69º.

Find B,B = 180º – (40º +75º) or 65º.

78. cot

SOLUTION:

Let u = , so u = .

So, cot = .

79. tan (arctan 3)

80. cos

SOLUTION:

Let u = cos– 1

and substitute.

82. cos – 1

SOLUTION:

exist, so cos – 1

does not exist.

83. sin

SOLUTION:

So, sin = sin u = .

x y ln y 14.1 19.5 2.97 18.0 26.4 3.27 18.3 29.7 3.39 18.7 32.3 3.48 19.1 34.4 3.54 19.4 36.2 3.59 19.6 37.7 3.63

85. A ⊂ B

86. D ⊂ U

A (x + 1)

B (x – 1)2

C 3x – 1 D 3x

E 3(x – 1)2

SOLUTION:

⋅ tan θ ?

SOLUTION:

is the length of ?

F 5 G 4 H 3.2

SOLUTION:

tan x:

cot x:

66. tan x

SOLUTION: cot x =

sec x:

cos x:

csc x:

sin x:

numbers.

SOLUTION:

69. tan2 + 1 = sec

SOLUTION:

tan2 + 1 = sec2

70. cot2 + 1 = csc

SOLUTION:

cos θ =

tan θ =

csc θ =

sec θ =

cot θ =

Quotient Identities

tan θ =

cot θ =

2 θ + cos

2 θ = 1

cot2 θ + 1 = csc

Odd-Even Identities

SOLUTION:

Find B, B = 180º – (122º +37º) or 21º.

Find C, C = 180º – (65º +46º) or 69º.

Find B,B = 180º – (40º +75º) or 65º.

78. cot

SOLUTION:

Let u = , so u = .

So, cot = .

79. tan (arctan 3)

80. cos

SOLUTION:

Let u = cos– 1

and substitute.

82. cos – 1

SOLUTION:

exist, so cos – 1

does not exist.

83. sin

SOLUTION:

So, sin = sin u = .

x y ln y 14.1 19.5 2.97 18.0 26.4 3.27 18.3 29.7 3.39 18.7 32.3 3.48 19.1 34.4 3.54 19.4 36.2 3.59 19.6 37.7 3.63

85. A ⊂ B

86. D ⊂ U

A (x + 1)

B (x – 1)2

C 3x – 1 D 3x

E 3(x – 1)2

SOLUTION:

⋅ tan θ ?

SOLUTION:

is the length of ?

F 5 G 4 H 3.2

SOLUTION:

tan x:

cot x:

66. tan x

SOLUTION: cot x =

sec x:

cos x:

csc x:

sin x:

numbers.

SOLUTION:

69. tan2 + 1 = sec

SOLUTION:

tan2 + 1 = sec2

70. cot2 + 1 = csc

SOLUTION:

cos θ =

tan θ =

csc θ =

sec θ =

cot θ =

Quotient Identities

tan θ =

cot θ =

2 θ + cos

2 θ = 1

cot2 θ + 1 = csc

Odd-Even Identities

SOLUTION:

Find B, B = 180º – (122º +37º) or 21º.

Find C, C = 180º – (65º +46º) or 69º.

Find B,B = 180º – (40º +75º) or 65º.

78. cot

SOLUTION:

Let u = , so u = .

So, cot = .

79. tan (arctan 3)

80. cos

SOLUTION:

Let u = cos– 1

and substitute.

82. cos – 1

SOLUTION:

exist, so cos – 1

does not exist.

83. sin

SOLUTION:

So, sin = sin u = .

x y ln y 14.1 19.5 2.97 18.0 26.4 3.27 18.3 29.7 3.39 18.7 32.3 3.48 19.1 34.4 3.54 19.4 36.2 3.59 19.6 37.7 3.63

85. A ⊂ B

86. D ⊂ U

A (x + 1)

B (x – 1)2

C 3x – 1 D 3x

E 3(x – 1)2

SOLUTION:

⋅ tan θ ?

SOLUTION:

is the length of ?

F 5 G 4 H 3.2

SOLUTION:

tan x:

cot x:

66. tan x

SOLUTION: cot x =

sec x:

cos x:

csc x:

sin x:

numbers.

SOLUTION:

69. tan2 + 1 = sec

SOLUTION:

tan2 + 1 = sec2

70. cot2 + 1 = csc

SOLUTION:

cos θ =

tan θ =

csc θ =

sec θ =

cot θ =

Quotient Identities

tan θ =

cot θ =

2 θ + cos

2 θ = 1

cot2 θ + 1 = csc

Odd-Even Identities

SOLUTION:

Find B, B = 180º – (122º +37º) or 21º.

Find C, C = 180º – (65º +46º) or 69º.

Find B,B = 180º – (40º +75º) or 65º.

78. cot

SOLUTION:

Let u = , so u = .

So, cot = .

79. tan (arctan 3)

80. cos

SOLUTION:

Let u = cos– 1

and substitute.

82. cos – 1

SOLUTION:

exist, so cos – 1

does not exist.

83. sin

SOLUTION:

So, sin = sin u = .

x y ln y 14.1 19.5 2.97 18.0 26.4 3.27 18.3 29.7 3.39 18.7 32.3 3.48 19.1 34.4 3.54 19.4 36.2 3.59 19.6 37.7 3.63

85. A ⊂ B

86. D ⊂ U

A (x + 1)

B (x – 1)2

C 3x – 1 D 3x

E 3(x – 1)2

SOLUTION:

⋅ tan θ ?

SOLUTION:

is the length of ?

F 5 G 4 H 3.2

SOLUTION:

5-1 Trigonometric Identities - Montville Township Public ...3/(5(35(6(17$7,216 In this problem, you...

Documents

Transcript of 5-1 Trigonometric Identities - Montville Township Public ...3/(5(35(6(17$7,216 In this problem, you...