What is the general solution for 3-3sin(x)=6cos^2(x) ?

The general solution for 3-3sin(x)=6cos^2(x) is x= pi/2+2pin,x=(7pi)/6+2pin,x=(11pi)/6+2pin

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3-3sin(x)=6cos^2(x)

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

x = \frac{π}{2} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Solution steps

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

Subtract

6 cos^{2} (x)

from both sides

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

Rewrite using trig identities

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

Solve by substitution

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

sin (x) = 1 : x = \frac{π}{2} + 2 πn

sin (x) = - \frac{1}{2} : x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

- 2 tan^{2} (θ) + 2 tan (θ) + 1 = 3 tan (θ)

cos (x) = \frac{60}{61}, tan (x) = \frac{11}{60}

sin (x) = - 2 sin (x)

19 cot^{2} (x) - 19 = 0

θ, 2 sin (θ) = csc (θ) + 1

What is the general solution for 3-3sin(x)=6cos^2(x) ?
The general solution for 3-3sin(x)=6cos^2(x) is x= pi/2+2pin,x=(7pi)/6+2pin,x=(11pi)/6+2pin