maths extended trigonometry. trigonometrical identities?

a) 2 sin x = cos (x + 60°) = cos x cos 60° - sin x sin 60° = 1/2 cos x - √3/2 sin x

(2 + √3/2) sin x = 1/2 cos x

tan x = 1/(4 + √3)

x ≈ 9.90° or 189.90°

b) cos (x + 45) = cos x

cos x cos 45° - sin x sin 45° = cos x

cos x (√2/2 - 1) = √2/2 sin x

(√2/2 - 1)/(√2/2) = tan x

1 - √2 = tan x

x = 157.5° or 337.5°

c) sin (x - 30°) = 1/2 cos x

sin x cos 30° - cos x sin 30° = 1/2 cos x

√3/2 sin x = cos x

tan x = 2/√3

x ≈ 49.1°, 229.1°

d) 3 sin (x + 10°) = 4 cos (x - 10°)

3 sin x cos 10° + 3 cos x sin 10° = 4 cos x cos 10° + 4 sin x sin 10°

(3 cos 10° - 4 sin 10°) sin x = (4 cos 10° - 3 sin 10°) cos x

tan x = (4 cos 10° - 3 sin 10°)/(3 cos 10° - 4 sin 10°)

tan x = 1.5126...

x ≈ 56.5°, 336.5°

a) 2sinx=cos(x+60)

=> 2 sin x = (1/2)cos x - ( √3/2) sin x

multiply with 2

=> 4 sin x + √3 sin x = cos x

sin x (4 + √3) = cos x

divide by cos x

tan x (4 + √3) = 1

tan x = 1 /(4 + √3) = (4 -√3) /13

x = 9.9° and 189.9 °

b) cos(x+45) = cosx

=> cos(x + 45) = cos(360 - x)

=> ( x + 45) = 360 - x

=> 2x = 315

x = 315/2 = 157.5°

c) sin(x-30)=1/2cosx

=>√3/2 sin x - 1/2 cos x = 1/2 cos x

=> √3 sin x = 2 cos x

divide by cos x

tan x = 2/√3

x = 49.11° and 229.11°

d) 3sin(x+10 ) = 4cos(x-10)

3(sin x cos 10 + cos x sin(10)) = 4( cos x cos 10 + sin x sin 10)

=> 2.9544 sin x + 0.5209 cos x = 3.939 cos x + 0.695 sin x

=>2.2594 sin x = 3.4181 cos x

=> tan x = 1.5128

x = 56.5349° and 236.5349°