What is the general solution for 2cos^2(θ)-3sin(θ)+2=0 ?

The general solution for 2cos^2(θ)-3sin(θ)+2=0 is θ=1.01746…+2pin,θ=pi-1.01746…+2pin

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2cos^2(θ)-3sin(θ)+2=0

2 cos^{2} (θ) - 3 sin (θ) + 2 = 0

θ = 1.01746 \dots + 2 πn, θ = π - 1.01746 \dots + 2 πn

Solution steps

2 cos^{2} (θ) - 3 sin (θ) + 2 = 0

Rewrite using trig identities

4 - 2 sin^{2} (θ) - 3 sin (θ) = 0

Solve by substitution

sin (θ) = - \frac{3 + 41}{4}, sin (θ) = \frac{41 - 3}{4}

sin (θ) = - \frac{3 + 41}{4} :

No Solution

sin (θ) = \frac{41 - 3}{4} : θ = arcsin (\frac{41 - 3}{4}) + 2 πn, θ = π - arcsin (\frac{41 - 3}{4}) + 2 πn

Combine all the solutions

θ = arcsin (\frac{41 - 3}{4}) + 2 πn, θ = π - arcsin (\frac{41 - 3}{4}) + 2 πn

Show solutions in decimal form

θ = 1.01746 \dots + 2 πn, θ = π - 1.01746 \dots + 2 πn

sin (x) + sin (2 x) + sin (3 x) = 0

sin^{2} (x) - cos^{2} (x) = 1 + cos (x)

36 cos^{2} (x) - 9 = 0

sin (x) = \frac{5}{6}

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

What is the general solution for 2cos^2(θ)-3sin(θ)+2=0 ?
The general solution for 2cos^2(θ)-3sin(θ)+2=0 is θ=1.01746…+2pin,θ=pi-1.01746…+2pin