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=2pin,x=pi+2pin,x= pi/3+2pin,x=(2pi)/3+2pin

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

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

x = 2 πn, x = π + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 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) = 0, sin (x) = \frac{3}{2}

sin (x) = 0 : x = 2 πn, x = π + 2 πn

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

Combine all the solutions

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

tan (2 θ) - 2 cos (θ) = 0

tan (θ) = \frac{12}{5}, π < θ < \frac{3 π}{2}

3 tan (3 θ) + 1 = 0, 0 \leq θ \leq 2 π

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

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

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=2pin,x=pi+2pin,x= pi/3+2pin,x=(2pi)/3+2pin