Section 5.5 Multiple-Angle and Product.Sum Formulas

[Pages:10]272 PART I: Solutions to Odd-Numbered Exercises and Practice Tests

87, x = O: y = -?(0 - 10) + 14 = 5 + 14 = 19. y-intercept: (0, 19) y = O: 0 = -g1(x- 10)? 14 = -?x + 19 ==> x = 38. x-intercept: (38,0)

89. x = o: 12(0) - 91 - 5 = 9 - 5 = 4. y-intercept: (0, 4) y = O: 12x - 91 = 5 ~ x = 7, 2. x-intercepts: (2, 0), (73 O)

91. arccos -~- = ~ because COS6 2~ = ~

93. arcsin 1 = -~ because sin ~ = 1.

Section 5.5 Multiple-Angle and Product.Sum Formulas

You should know the following double-angle formulas.

(a) sin 2u = 2 sin u cos u (b) cos 2u = cos2 u - sin2 u

= 2 cos2 u - 1 = 1 - 2 sin2 u

2 tan u (c) tan 2u - 1 - tan2 u [] You should be able to reduce the power of a trigonometric function.

(a)

sinz

u

= 1

-

cos2u 2

(b) cos21u+=cos22u

[] You should be able to use the half-angle formulas.

(c)

tanz

u11

=+- cc-oo-ss

2u 2u

(a) siznu~~l=-c+?suzu~2/l+c?su(b) cos==+

[] You should be able to use the product-sum formulas. (a) sin u sin v =~1 [cos(u - v) - cos(u + v)]

2

(c)

tan2u -1

-

cos sin u

u

-

sin u 1 + cos u

1 (b) cos u cos v = ~ [cos(u - v) + cos(u + v)]

(c) sin u cos v =1~ [sin(u + v) + sin(u - v)] [] You should be able to use the sum-product formulas.

! (d) cos u sin v = ~ [sin(u + v) - sin(u - v)]

(a) sinx+siny 2sin x+y x-y

x+y x-y (b) sinx-siny = 2 cos(---~)sin(--~)

(c) cos x + cos y = 2 cos~--~)cos --~

(d) cosx-cosy=-2sinX+Y sin x-y

273 PART I: Solutions to Odd-Numbered Exercises and Practice Tests Solutions to Odd-Numbered Exercises

Figure for Exercises 1-7

1. sin 0 = ~

5.

tan20

-

2 tan 0 1 - tan2

0

--. 2(3/4) 1 - (3/4)2

3/2

1 - (9/16)

3 16

2 7

24

9. Solutions: O, 1.047, 3.142, 5.236

sin 2x - sin x = 0

2 sin x cos x - sin x = 0

sin x(2 cos x - 1) = 0

sin3c=O or 2cosx- 1 =0

X-" O,'rr

1 cosx =--2

? r 5~r x = O, ~, "n', 3

11. Solutions: 0.1263, 1.4445, 3.2679, 4.5860

sin 0 = ~

COS 0 = ~

tan 0 = ~

3. cos 20=2cos20- 1

= 2(~)~- 1

32 25 -- 25 25

7 -- 25

7. csc 20

-

1 sin 20

--.

1

.2sin0cos0

..._

1

2(3/5)(4/5)

25

24

274 PART I: Solutions to Odd-Numbered Exercises and Practice Tests

13. Solutions: 1.047, 3.142, 5.236

cos 2x= -cos x 2cos2x- 1 =-cosx

2cos2x+cosx- 1=0

(2 cos x - 1)(cos x + 1) = 0

2cosx= 1 or cosx=-I

1 COS X = -2-

X = ~

,tr 5"tr 3'3

15. Solutions: O, 1.571, 3.142, 4.712

sin 4x = -2 sin 2x

sin4x + 2 sin2x = 0

2 sin 2x cos 2x + 2 sin 2x = 0

2 sin 2x(cos 2x + 1) = 0

2sin2x=O or cos2x+ 1=0

sin 2x = 0

cos 2x = -1

2x = nqr x = ~'/nr

2x = ?r + 2n~r

x = -~ + nq~"

7r

"," x=O,-~,

,rr, 32"n"

-tr 3,0" x-2' 2

17. 8 sin x cos x = 4(2 sin x cos x) = 4 sin 2x

19. 5 - 10 sin2 x = 5(1 - 2 sin2 x) = 5 cos 2x

21.

sinu

=3~.

O ................
................

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