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

The general solution for 3+3cos(x)=2sin^2(x) is x=(2pi)/3+2pin,x=(4pi)/3+2pin,x=pi+2pin

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

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

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

Solution steps

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

Subtract

2 sin^{2} (x)

from both sides

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

Rewrite using trig identities

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

Solve by substitution

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

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

cos (x) = - 1 : x = π + 2 πn

Combine all the solutions

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

cos (θ) = - sin (θ)

2 csc (x) + 2 = 4

sec^{2} (θ) + 5 sec (θ) - 6 = 0

cos (x) = 0.75

cos (x) = 0.76

What is the general solution for 3+3cos(x)=2sin^2(x) ?
The general solution for 3+3cos(x)=2sin^2(x) is x=(2pi)/3+2pin,x=(4pi)/3+2pin,x=pi+2pin