What is the general solution for cos(2x)-sqrt(3)sin(x)=1 ?

The general solution for cos(2x)-sqrt(3)sin(x)=1 is x=(4pi)/3+2pin,x=(5pi)/3+2pin,x=2pin,x=pi+2pin

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cos(2x)-sqrt(3)sin(x)=1

cos (2 x) - 3 sin (x) = 1

x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = 2 πn, x = π + 2 πn

Solution steps

cos (2 x) - 3 sin (x) = 1

Subtract

1

from both sides

cos (2 x) - 3 sin (x) - 1 = 0

Rewrite using trig identities

- 2 sin^{2} (x) - sin (x) 3 = 0

Solve by substitution

sin (x) = - \frac{3}{2}, sin (x) = 0

sin (x) = - \frac{3}{2} : x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

sin (x) = 0 : x = 2 πn, x = π + 2 πn

Combine all the solutions

x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = 2 πn, x = π + 2 πn

- 2 sin (x) - sin (2 x) = 0

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

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

tan (θ) = - 4

cos (θ) = - \frac{1}{3}

What is the general solution for cos(2x)-sqrt(3)sin(x)=1 ?
The general solution for cos(2x)-sqrt(3)sin(x)=1 is x=(4pi)/3+2pin,x=(5pi)/3+2pin,x=2pin,x=pi+2pin