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

The general solution for sin(x)-sqrt(3)cos(x)=2 is x=(5pi)/6+2pin

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

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

x = \frac{5 π}{6} + 2 πn

Solution steps

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

Add

3 cos (x)

to both sides

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

Square both sides

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

Subtract

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

from both sides

sin^{2} (x) - 4 - 43 cos (x) - 3 cos^{2} (x) = 0

Rewrite using trig identities

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

Solve by substitution

cos (x) = - \frac{3}{2}

cos (x) = - \frac{3}{2} : x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

Combine all the solutions

x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

Verify solutions by plugging them into the original equation

x = \frac{5 π}{6} + 2 πn

3 tan (15 x^{5}) + 1 = 0

4 cos (θ) - 3 sec (θ) = 0

sin (x) = \frac{π}{6}

(2 sin (4 x) - 1) \cdot (1 + tan (x)) = 0

arctan (θ) = 3

What is the general solution for sin(x)-sqrt(3)cos(x)=2 ?
The general solution for sin(x)-sqrt(3)cos(x)=2 is x=(5pi)/6+2pin