What is the general solution for -4sin^2(θ)-7sin(θ)+4=0 ?

The general solution for -4sin^2(θ)-7sin(θ)+4=0 is θ=0.47098…+2pin,θ=pi-0.47098…+2pin

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-4sin^2(θ)-7sin(θ)+4=0

- 4 sin^{2} (θ) - 7 sin (θ) + 4 = 0

θ = 0.47098 \dots + 2 πn, θ = π - 0.47098 \dots + 2 πn

Solution steps

- 4 sin^{2} (θ) - 7 sin (θ) + 4 = 0

Solve by substitution

sin (θ) = - \frac{7 + 113}{8}, sin (θ) = \frac{113 - 7}{8}

sin (θ) = - \frac{7 + 113}{8} :

No Solution

sin (θ) = \frac{113 - 7}{8} : θ = arcsin (\frac{113 - 7}{8}) + 2 πn, θ = π - arcsin (\frac{113 - 7}{8}) + 2 πn

Combine all the solutions

θ = arcsin (\frac{113 - 7}{8}) + 2 πn, θ = π - arcsin (\frac{113 - 7}{8}) + 2 πn

Show solutions in decimal form

θ = 0.47098 \dots + 2 πn, θ = π - 0.47098 \dots + 2 πn

cos (8 x) - cos (4 x) = 0

cos (x) = \frac{4}{3}

cos^{2} (θ) - sin (θ) cos (θ) = 0

x, cos (2 x) + cos (x) = 0

2 sin (θ) + cos (2 θ) = 1

What is the general solution for -4sin^2(θ)-7sin(θ)+4=0 ?
The general solution for -4sin^2(θ)-7sin(θ)+4=0 is θ=0.47098…+2pin,θ=pi-0.47098…+2pin