What is the general solution for 4cos^2(θ)=3+3sin(θ) ?

The general solution for 4cos^2(θ)=3+3sin(θ) is θ=(3pi)/2+2pin,θ=0.25268…+2pin,θ=pi-0.25268…+2pin

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4cos^2(θ)=3+3sin(θ)

4 cos^{2} (θ) = 3 + 3 sin (θ)

θ = \frac{3 π}{2} + 2 πn, θ = 0.25268 \dots + 2 πn, θ = π - 0.25268 \dots + 2 πn

Solution steps

4 cos^{2} (θ) = 3 + 3 sin (θ)

Subtract

3 + 3 sin (θ)

from both sides

4 cos^{2} (θ) - 3 - 3 sin (θ) = 0

Rewrite using trig identities

1 - 3 sin (θ) - 4 sin^{2} (θ) = 0

Solve by substitution

sin (θ) = - 1, sin (θ) = \frac{1}{4}

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

sin (θ) = \frac{1}{4} : θ = arcsin (\frac{1}{4}) + 2 πn, θ = π - arcsin (\frac{1}{4}) + 2 πn

Combine all the solutions

θ = \frac{3 π}{2} + 2 πn, θ = arcsin (\frac{1}{4}) + 2 πn, θ = π - arcsin (\frac{1}{4}) + 2 πn

Show solutions in decimal form

θ = \frac{3 π}{2} + 2 πn, θ = 0.25268 \dots + 2 πn, θ = π - 0.25268 \dots + 2 πn

4 cos (3 θ) + 23 = 0

cos (2 θ) + 3 cos (θ) - 1 = 0

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

2 sin (x) + tan (x) = 0

arctan (x + 1) + arctan (x - 1) = arctan (12)

What is the general solution for 4cos^2(θ)=3+3sin(θ) ?
The general solution for 4cos^2(θ)=3+3sin(θ) is θ=(3pi)/2+2pin,θ=0.25268…+2pin,θ=pi-0.25268…+2pin