6sin^2(θ)-5sin(θ)=-1

Solution

6 sin^{2} (θ) - 5 sin (θ) = - 1

θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn, θ = 0.33983 \dots + 2 πn, θ = π - 0.33983 \dots + 2 πn

Solution steps

6 sin^{2} (θ) - 5 sin (θ) = - 1

Solve by substitution

sin (θ) = \frac{1}{2}, sin (θ) = \frac{1}{3}

sin (θ) = \frac{1}{2} : θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn

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

Combine all the solutions

θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn, θ = arcsin (\frac{1}{3}) + 2 πn, θ = π - arcsin (\frac{1}{3}) + 2 πn

Show solutions in decimal form

θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn, θ = 0.33983 \dots + 2 πn, θ = π - 0.33983 \dots + 2 πn

csc (x) = - 4

tan (x) = - 3, - 27 0^{\circ} \leq x \leq 27 0^{\circ}

5 cos^{2} (θ) = - 13 cos (θ) + 6

- 3 sin (x) - cos (x) = 0

tan (θ) + sec (θ) = 1