o TRIGONOMETRY Answers - Worksheet E I

1 a sin(A+B)==sinAcosB+cosAsinB (1)sin (A - B) ==sin A cos B - cosA sin B (2)

b (1) + (2) sin (A + B) + sin (A - B) ==sin A cos B+ cosA sin B + sin A cos B - cosA sin B~ 2 sin A cos B ==sin (A + B) + sin (A - B)

c cos(A+B)==cosAcosB-sinAsinB (3)cos (A - B) ==cos A cos B + sin A sin B (4)

(3) + (4) 2 cos A cos B ==cos (A + B) + cos (A - B)(4)-(3) 2sinAsinB==cos(A-B)-cos(A+B)

2 a = sin (30 + 10)° + sin (30 - 10)°= sin 40° + sin 20°

b = cos (36 + 18)° + cos (36 - 18)°= cos 54° + cos 18°

c = Hsin (49 + 25t - sin (49 - 25)°]

= .isin 74° - .isin 24°2 2

d = cos (3A - A) - cos (3A + A)

= cos 2A - cos 4A

e = sin (5A + 2A) - sin (5A - 2A)= sin 7A - sin 3A

f = 2[cos (3A + B) + cos (3A - B)]= 2 cos (3A + B) + 2 cos (3A - B)

h = sin [A + (A + 40°)] - sin [A - (A + 40°)]

= sin (2A + 40°) - sin (-40°)

= sin (2A + 40°) + sin 40°

g = Hsin (A + 6B) + sin (A - 6B)]

= tsin (A + 6B) + tsin (A - 6B)

3 a 2 sin A cos B ==sin (A + B) + sin (A - B)let P=A+B (1) and Q=A-B (2)

(1)+(2) ~ 2A=P+Q ~ A= P+Q, (1)-(2) ~ 2B=P-Q ~ B= P-Q2 2

:. sinP+sinQ==2sin P+Q cos P-Q2 2

b let P = A + B and Q = A - B in each part

2 cos A sin B ==sin (A + B) - sin (A - B) ~ sin P - sin Q ==2 cos P + Q sin P - Q2 2ii 2 cos A cos B ==cos (A + B) + cos (A - B) ~ cos P + cos Q ==2 cos P + Q cos P - Q

2 2iii 2sinAsinB==cos(A-B)-cos(A+B) ~ cosQ-cosP==2sin P+Q sin P-Q

2 2~ cos P - cos Q ==-2 sin P+Q sin P-Q

2 2

4 2 25+15 25-15a = cos -2- COS-2-

= 2 cos 20° cos 5°

b 2 84+30. 84-30= COS-2- Sill-2-

= 2 cos 57° sin 27°

2· 5A+A 5A-AC = Sill-2- COS-2-

= 2 sin 3A cos 2A

d 2· A+2A • A-2A=- Sill-2- Sill-2-

= -2 sin 3A sin (_A) = 2 sin 3A sin A2 2 2 2

2. 2A+4B • 2A-4Be =- sin -2- Sill-2-

= -2 sin (A + 2B) sin (A - 2B)

f = 2 sin 2A;90 cos (-~O )

=2sin(A +45)cos(-15)=2sin(A +45°) cos 15°

g = 4 COSA+3A COSA-3A2 2

= 4 cos 2A cos (-A) = 4 cos 2A cos A

2 4A+B· 3B-2Ah = COS-2- sm -2-

= 2 cos (2A + t B) sin ( tB - A)

I: TRIGONOMETRY Answers - Worksheet E page 21

5 a 2 cos ~ sin ~ = 02 2cos 2x sin x = 0

cos 2x = 0 or sin x = 0

2x = ~ 21t - ~ or x = 0 1t2' 2 '

2x= ~ 3n: or x=O 1t2' 2 '

X - 0 n: 3rr 1t- , 4'4'

c cos (x - 5x) - cos (x + 5x) = cos 4x

cos (-4x) - cos 6x = cos 4x

cos 4x - cos 6x = cos 4x

cos6x=0

6x = 1-, 21t - 1-, 21t + 1-,41t - ~ 41t + ~ 61t - ~

2' 2' 2_ 1t 3rr 5rr 7rr 9rr l l n-2'2'2'2'2'2

_ rr rr 51t 7rr 3rr llrrx- IT' 4' IT' IT' 4' 12

x+..!. x-..!.e 2 sin -T- cos -T- = 0

sin 1.x cos .Lx = 04 4

sin 1.x = 0 or cos .1x = 04 4tx = 0 or (none in interval)

x=O

6 a cos (2x + 3x) + cos (2x - 3x) - cos x = 0

cos 5x + cos (-x) - cosx = 0

cos 5x + cos x - cos x = 0

cos 5x =0

5x = 90, 360 - 90, 360 + 90,

720 - 90, 720 + 90= 90, 270, 450, 630, 810

x = 18°,54°,90°, 126°, 162°

b cos 4x - cos x = 0

-2 sin 4x+x sin 4x-x = 02 2sin 2.xsin 1.x=O

2 2

sin 2.x = 0 or sin 1.x = 02 2

fx = 0, 1t,21t or sin tx = 0, 1t

X = 0 2rr 2rr 4rr, 5 ' 3 ' 5

d 4[sin (2x + 1-) - sin ~] = 1

sin (2x + ~) - .L = .1224sin (2x + 1-) = t2x + 1- = 1t- 0.8481, 21t + 0.8481

= 2.2935, 7.1312

2x = 0.7227,5.5605

x = 0.36, 2.78

2 3x+x 3x-x 2xf cos -2- cos -2- = COS

2 cos 2x cos x = cos 2x

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

cos 2x = 0 or cos x = 12x = 1-, 21t - 1- or x = ~

2x = ~ 3rr or x = ~2' 2 3

x= ~ ~ 3rr4' 3' 4

b 2cos 3x+2x sin ~ =02 2COS2.x sin .Lx = 0

2 2

COS2.x = 0 or sin .Lx = 02 2fx=90,360-90,360+90 or 1x=0

2.x = 90 270 450 or .Lx = 02 " 2

X = 0,36°, 108°, 180°

1- TRIGONOMETRY Answers - Worksheet E page 3 1

2· 4x+Zx 4x-Zx • 3C Sin -Z- COS-Z- = Sin X

2 sin 3x cos x = sin 3xsin 3x(2 cos x-I) = 0

sin 3x = 0 or cos x = t

3x = 0, 180,360,540 or x = 60x = 0, 60°,120°, 180°

e Hsin (5x +x) - sin (5x - x)] + sin 4x = 0

tsin 6x - tsin 4x + sin 4x = 0.1sin 6x + .1sin 4x = 0Z Zsin 6x+4x cos 6x-4x = 0

Z Zsin 5x cos x = 0sin 5x = 0 or cos x = 05x = 0, 180,360,540,720,900 or x = 90x = 0, 36°, 72°, 90°, 108°, 144°, 180°

7 LHS 2· x+3x x-3x • 2a. = SIn -z- cos -z- + Sin X

= 2 sin 2x cos (-x) + sin 2x

= 2 sin 2x cos x + sin 2x

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

d cos 2x - cos (x - 60) = 0-2 sin 3x-6O sin x+60 = 0

2 zsin (tx - 30) sin (tx + 30) = 0sin ( t x - 30) = 0 or sin ( t x + 30) = 0tx - 30 = 0, 180 or (none in interval)l.x = 30 210z 'x= 20°,140°

f 2 sin x+3x COSx-3x = 2 COSx+3x COSx-3xZ Z Z z

sin 2x cos (-x) = cos 2x cos (-x)sin 2x cos x = cos 2x cos x

cos x(sin 2x - cos 2x) = 0cos x = 0 or sin 2x = cos 2xcos x = 0 or tan 2x = 1x = 90 or 2x = 45, 180 + 45 = 45, 225x = 22.5°, 90°, 112.5°

2 . x+3x • x-3xb LHS = - sm-z-sm-z-

2cos x+3x COSx-3x2 Z

= -sin2xsin(-x)cos2xcos(-x)sin2xsinxcos2xcosx

= tan x tan 2x=RHS