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

The general solution for sqrt(3)cos(x)=1+sin(x) is x=(3pi)/2+2pin,x= pi/6+2pin

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

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

x = \frac{3 π}{2} + 2 πn, x = \frac{π}{6} + 2 πn

Solution steps

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

Square both sides

(3 cos (x))^{2} = (1 + sin (x))^{2}

Subtract

(1 + sin (x))^{2}

from both sides

3 cos^{2} (x) - 1 - 2 sin (x) - sin^{2} (x) = 0

Rewrite using trig identities

2 - 2 sin (x) - 4 sin^{2} (x) = 0

Solve by substitution

sin (x) = - 1, sin (x) = \frac{1}{2}

sin (x) = - 1 : x = \frac{3 π}{2} + 2 πn

sin (x) = \frac{1}{2} : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Combine all the solutions

x = \frac{3 π}{2} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Verify solutions by plugging them into the original equation

x = \frac{3 π}{2} + 2 πn, x = \frac{π}{6} + 2 πn

6 cos^{2} (\frac{x}{2}) + cos (x) + 1 = 0

- 2 sin (θ) = 1

cos (θ) = - \frac{7}{3}

0 = - 6 sin (x) + 6 cos (2 x)

tan (2 x + 1) = \frac{1}{2}

What is the general solution for sqrt(3)cos(x)=1+sin(x) ?
The general solution for sqrt(3)cos(x)=1+sin(x) is x=(3pi)/2+2pin,x= pi/6+2pin