6sin^2(θ)-sin(θ)-2=0

Solution

6 sin^{2} (θ) - sin (θ) - 2 = 0

θ = 0.72972 \dots + 2 πn, θ = π - 0.72972 \dots + 2 πn, θ = \frac{7 π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn

Solution steps

6 sin^{2} (θ) - sin (θ) - 2 = 0

Solve by substitution

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

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

sin (θ) = - \frac{1}{2} : θ = \frac{7 π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn

Combine all the solutions

θ = arcsin (\frac{2}{3}) + 2 πn, θ = π - arcsin (\frac{2}{3}) + 2 πn, θ = \frac{7 π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn

Show solutions in decimal form

θ = 0.72972 \dots + 2 πn, θ = π - 0.72972 \dots + 2 πn, θ = \frac{7 π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn

tan (x) sin (x) + sin (x) = 0

2 sin (\frac{x}{5}) + 3 = 0

sin (\frac{θ}{2}) = 1

x, z = arctan (\frac{y}{x})

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