12sin^2(θ)-6sin(θ)=5

Solution

12 sin^{2} (θ) - 6 sin (θ) = 5

θ = 1.22919 \dots + 2 πn, θ = π - 1.22919 \dots + 2 πn, θ = - 0.45807 \dots + 2 πn, θ = π + 0.45807 \dots + 2 πn

Solution steps

12 sin^{2} (θ) - 6 sin (θ) = 5

Solve by substitution

sin (θ) = \frac{3 + 69}{12}, sin (θ) = \frac{3 - 69}{12}

sin (θ) = \frac{3 + 69}{12} : θ = arcsin (\frac{3 + 69}{12}) + 2 πn, θ = π - arcsin (\frac{3 + 69}{12}) + 2 πn

sin (θ) = \frac{3 - 69}{12} : θ = arcsin (\frac{3 - 69}{12}) + 2 πn, θ = π + arcsin (- \frac{3 - 69}{12}) + 2 πn

Combine all the solutions

θ = arcsin (\frac{3 + 69}{12}) + 2 πn, θ = π - arcsin (\frac{3 + 69}{12}) + 2 πn, θ = arcsin (\frac{3 - 69}{12}) + 2 πn, θ = π + arcsin (- \frac{3 - 69}{12}) + 2 πn

Show solutions in decimal form

θ = 1.22919 \dots + 2 πn, θ = π - 1.22919 \dots + 2 πn, θ = - 0.45807 \dots + 2 πn, θ = π + 0.45807 \dots + 2 πn

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

cos (c) = 0.87

8 sin^{2} (x) - 2 sin (x) - 1 = 0

sin (\frac{1}{4} x) = 0

4 cos^{2} (x) - 4 cos (x) = - 1