(tan(θ)-2)(16sin^2(θ)-1)=0

Solution

(tan (θ) - 2) (16 sin^{2} (θ) - 1) = 0

θ = 1.10714 \dots + πn, θ = 0.25268 \dots + 2 πn, θ = π - 0.25268 \dots + 2 πn, θ = - 0.25268 \dots + 2 πn, θ = π + 0.25268 \dots + 2 πn

Solution steps

(tan (θ) - 2) (16 sin^{2} (θ) - 1) = 0

Solving each part separately

tan (θ) - 2 = 0 or 16 sin^{2} (θ) - 1 = 0

tan (θ) - 2 = 0 : θ = arctan (2) + πn

16 sin^{2} (θ) - 1 = 0 : θ = arcsin (\frac{1}{4}) + 2 πn, θ = π - arcsin (\frac{1}{4}) + 2 πn, θ = arcsin (- \frac{1}{4}) + 2 πn, θ = π + arcsin (\frac{1}{4}) + 2 πn

Combine all the solutions

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

Show solutions in decimal form

θ = 1.10714 \dots + πn, θ = 0.25268 \dots + 2 πn, θ = π - 0.25268 \dots + 2 πn, θ = - 0.25268 \dots + 2 πn, θ = π + 0.25268 \dots + 2 πn

csc (θ) = 1.9

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

sin (x) = 2 cos (\frac{x}{2})

2 sin (x) + \frac{1}{sin ( x )} = 3

cos (x - \frac{π}{4}) = \frac{3}{2}