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

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

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

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

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

Solution steps

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

Add

3 cos (x)

to both sides

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

Square both sides

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

Subtract

(3 + 3 cos (x))^{2}

from both sides

sin^{2} (x) - 3 - 6 cos (x) - 3 cos^{2} (x) = 0

Rewrite using trig identities

- 2 - 4 cos^{2} (x) - 6 cos (x) = 0

Solve by substitution

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

cos (x) = - 1 : x = π + 2 πn

cos (x) = - \frac{1}{2} : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

Combine all the solutions

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

Verify solutions by plugging them into the original equation

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

cot^{2} (x) - 2 csc (x) = - 1

3 csc (x) + csc (x) cot (x) = 0

- 3 cos^{2} (θ) - 10 cos (θ) = - 3 cos (θ) - 6

- 4 sin (x) = - cos^{2} (x) + 4

sin^{2} (x) - 4 sin (x) + 3 = 0

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