CHAPTER 5 Analytic Trigonometry - KHSPreCalc

CHAPTER 5 Analytic Trigonometry

Section 5.1 Using Fundamental Identities ............................................................397 Section 5.2 Verifying Trigonometric Identities....................................................404 Section 5.3 Solving Trigonometric Equations......................................................410 Section 5.4 Sum and Difference Formulas ...........................................................425 Section 5.5 Multiple-Angle and Product-to-Sum Formulas ................................441 Review Exercises ........................................................................................................453 Problem Solving .........................................................................................................464 Practice Test .............................................................................................................471

? 2018 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

CHAPTER 5 Analytic Trigonometry

Section 5.1 Using Fundamental Identities

1. tan u 2. csc u

3. cot u

4. csc u

5. 1 6. -sin u

7. sec x = - 52, tan x < 0 x is in Quadrant II.

cos x

=

1 sec x

=

1

-

5 2

=

-

2 5

sin x =

1

-

-

2 5

2

=

1-

4 25

=

21 5

21

tan x

=

sin x cos x

=

5

-

2 5

=-

21 2

csc x

=

1 sin x

=

5 21

=

5 21 21

cot x

=

1 tan x

=

-

2 21

=

-

2

21 21

8. csc x = - 76, tan x > 0 x is in Quadrant III.

sin x

=

1 csc x

=

1

-

7 6

=

-

6 7

cos x = -

1

-

-

6 7

2

=

-

1-

36 49

=

-

13 7

tan x

=

sin x cos x

=

-

6 7

-

13 7

=

6 13

=

6 13 13

sec x

=

1 cos x

=

-

1 13 7

=-

7 13

=

-

7

13 13

cot x

=

1 tan x

=

1 6

=

13 6

13

9. sin = - 34, cos > 0 is in Quadrant IV.

cos =

1

-

-

3 4

2

=

1

-

9 16

=

7 4

tan

=

sin cos

=

-

3 4

7

=-

3 7

= -377

4

sec

=

1 cos

=

1= 7

4 7

=

47 7

4

cot

=

1 tan

=

1

-

3 7

=-

7 3

csc

=

1 sin

=

1

-

3 4

=

-

4 3

10.

cos

=

2, 3

sin

<

0

is in Quadrant IV.

sin = -

1

-

2 2 3

=

-

1-

4 9

=

-

5 3

tan

=

sin cos

=

-

5 2

2

=

-

5 2

3

sec

=

1 cos

=

1 2

=

3 2

3

cot

=

1 tan

=

1

-

5 2

=

-

2 5

=

-255

csc

=

1 sin

=

1

-

5 3

=-

3 5

= -355

? 2018 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 397

398 Chapter 5 Analytic Trigonometry

11. tan x = 23, cos x > 0 x is in Quadrant I.

cot x

=

1 tan x

=

1 2

=

3 2

3

sec x =

1

+

2 2 3

=

1+

4 9

=

13 3

csc x =

1

+

3 2 2

=

1+

9 4

=

13 2

sin x

=

1 csc x

=

1 13

=

2 13

=

2 13 13

2

cos x

=

1 sec x

=

1 13

=

3 13

=

3 13 13

3

cot x

=

1 tan x

=

1 2

=

3 2

3

12.

cot x

=

7, sin x 4

<

0

x is in Quadrant III.

tan x

=

1 cot x

=

1 7

=

4 7

4

sec x = -

1

+

42 7

=

-

1+

16 49

=

-

65 7

csc x = -

1

+

7 2 4

=

-

1+

49 16

=

-

65 4

sin x =

1 csc x

=

-

1 65 4

=-

4 65

=

-

4

65 65

cos x =

1 sec x

=

-

1 65 7

=-

7 65

=

-

7

65 65

13.

sec x cos x

=

1

cos

x

cos x

= 1

Matches (c).

( ) 14. cot2 x - csc2 x = csc2 x - 1 - csc2 x

= -1

Matches (b).

( ) ( ) 15. cos x 1 + tan2 x = cos x sec2 x

=

cos

x

1 cos2

x

1

= cos x

= sec x

Matches (f).

16.

cot

x sec x

=

cos x sin x

1 cos x

=

1 sin x

=

csc x

Matches (a).

17.

sec2 x sin2

- x

1

=

tan2 x sin2 x

=

sin2 x 1 cos2 x sin2 x

= sec2 x

Matches (e).

18.

cos2 ( 2) - x

cos x

=

sin2 x cos x

=

sin x sin cos x

x

=

tan

x

sin

x

Matches (d).

19.

tan cot sec

=

tan

1 tan

1

cos

=

1 1

cos = cos

20.

cos

2

-

x sec x

=

sin x sec x

=

sin

x

1 cos

x

= tan x

( ) 21. tan2 x - tan2 x sin2 x = tan2 x 1 - sin2 x

= tan2 x cos2 x

=

sin2 x cos2 x

cos2

x

= sin2 x

( ) 22. sin2 x sec2 x - sin2 x = sin2 x sec2 x - 1

= sin2 x tan2 x

23.

sec2 x - 1 sec x - 1

=

(sec x + 1)(sec x - 1)

sec x - 1

= sec x + 1

24.

cos x - 2 cos2 x - 4

=

cos x - 2

(cos x + 2)(cos x - 2)

=

1 cos x +

2

? 2018 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Section 5.1 Using Fundamental Identities 399

( ) 25. 1 - 2 cos2 x + cos4 x = 1 - cos2 x 2 ( ) = sin2 x 2

= sin4 x

( )( ) 26. sec4 x - tan4 x = sec2 x + tan2 x sec2 x - tan2 x = (sec2 x + tan2 x)(1)

= sec2 x + tan2 x

27. cot3 x + cot2 x + cot x + 1 = cot2 x(cot x + 1) + (cot x + 1)

= (cot x + 1)(cot2 x + 1)

= (cot x + 1)csc2 x

28. sec3 x - sec2 x - sec x + 1 = sec2 x(sec x - 1) - (sec x - 1)

= (sec2 x - 1)(sec x - 1)

= tan2 x(sec x - 1)

29. 3 sin2 x - 5 sin x - 2 = (3 sin x + 1)(sin x - 2) 30. 6 cos2 x + 5 cos x - 6 = (3 cos x - 2)(2 cos x + 3)

38.

cot u sin u + tan u cos u

=

cos sin

u u

(sin

u)

+

sin cos

u u

(cos

u)

= cos u + sin u

( ) 31. cot2 x + csc x - 1 = csc2 x - 1 + csc x - 1

= csc2 x + csc x - 2

= (csc x - 1)(csc x + 2)

( ) 32. sin2 x + 3 cos x + 3 = 1 - cos2 x + 3 cos x + 3

= -cos2 x + 3 cos x + 4

( ) = - cos2 x - 3 cos x - 4

= -(cos x + 1)(cos x - 4)

33.

tan csc

=

sin cos

1 sin

=

1 cos

= sec

( ) 39.

1 - sin2 x csc2 x - 1

=

cos2 x cot2 x

=

cos2 x tan2 x

=

cos2 x

sin2 x cos2 x

= sin2 x

40.

cos2 y 1 - sin y

=

1 - sin2 y 1 - sin y

=

(1 +

sin y)(1 - sin

1 - sin y

y)

=

1+

sin

y

41. (sin x + cos x)2 = sin2 x + 2 sin x cos x + cos2 x

( ) = sin2 x + cos2 x + 2 sin x cos x

= 1 + 2 sin x cos x

34. tan(-x) cos x = -tan x cos x

=

-

sin cos

x x

cos

x

= -sin x

42. (2 csc x + 2)(2 csc x - 2) = 4 csc2 x - 4

= (4 csc2 x - 1)

= 4 cot2 x

35.

sin (csc

- sin )

=

(sin

)

1 sin

- sin2

= 1 - sin2 = cos2

36.

cos x(sec x - cos x)

=

cos

x

1 cos

x

- cos x

= 1 - cos2 x

= sin2 x

37.

sin

tan

+

cos

=

(sin

)

sin cos

+

cos

=

sin2 cos

+

cos2 cos

=

sin2 + cos2 cos

=

1 cos

= sec

43.

1 1 + cos x

+

1 1 - cos x

=

1 - cos x + 1 + cos x

(1 + cos x)(1 - cos x)

2 = 1 - cos2 x

=

2 sin 2

x

= 2 csc2 x

44.

sec

1 x

+

1

-

1 sec x - 1

=

sec x - 1 - (sec x + 1) (sec x + 1)(sec x - 1)

=

sec x - 1 - sec x sec2 x - 1

-1

=

-2 tan2 x

=

-2

1 tan 2

x

= -2 cot2 x

? 2018 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

400 Chapter 5 Analytic Trigonometry

45.

cos x 1 + sin x

-

1

cos x - sin

x

=

cos x(1 - sin x) - cos x(1 + (1 + sin x)(1 - sin x)

sin x)

=

cos x

-

sin x cos x - cos x - sin

(1 + sin x)(1 - sin x)

x cos x

=

-2 sin x cos x 1 - sin2 x

=

-2 sin x cos x cos2 x

=

-2 sin x cos x

= -2 tan x

46.

sin x 1 + cos x

+

sin x 1 - cos x

=

sin x(1 - cos x) + sin x(1 + cos x) (1 + cos x)(1 - cos x)

=

sin

x

-

sin x cos x + sin x + sin

(1 + cos x)(1 - cos x)

x cos

x

=

1

2 sin x - cos2

x

2 sin x = sin2 x

=

2 sin x

= 2 csc x

47.

tan

x

-

sec2 x tan x

=

tan2 x - sec2 x tan x

=

-1 tan x

=

-cot x

48.

cos x 1 + sin x

+

1

+ sin cos x

x

=

cos2 x + (1 + sin x)2 cos x(1 + sin x)

=

cos2

x + 1 + 2 sin x +

cos x(1 + sin x)

sin 2

x

=

2 + 2 sin x

cos x(1 + sin x)

=

2(1 + sin x) cos x(1 + sin x)

2 = cos x

= 2 sec x

49.

sin2 y 1 - cos y

=

1 - cos2 y 1 - cos y

=

(1 +

cos y)(1 - cos

1 - cos y

y)

=

1+

cos

y

50.

tan

x

5 +

sec

x

tan tan

x x

- -

sec sec

x x

=

5(tan x - sec x)

tan2 x - sec2 x

=

5(tan

x- -1

sec

x)

= 5(sec x - tan x)

51.

y1

=

tan x + 1 sec x + csc x

=

sin x cos x

+

1

1 cos

x

+

1 sin x

sin x + cos x

=

cos x sin x + cos x

sin x cos x

=

sin

x+ cos

cos x

x sin x cos x

sin

x

+

cos

x

= sin x

2

- 2

2

-2

? 2018 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

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