-2sin(θ)+3cos(2θ)=0

Solution

- 2 sin (θ) + 3 cos (2 θ) = 0

θ = - 1.10430 \dots + 2 πn, θ = π + 1.10430 \dots + 2 πn, θ = 0.59416 \dots + 2 πn, θ = π - 0.59416 \dots + 2 πn

Solution steps

- 2 sin (θ) + 3 cos (2 θ) = 0

Rewrite using trig identities

(1 - 2 sin^{2} (θ)) \cdot 3 - 2 sin (θ) = 0

Solve by substitution

sin (θ) = - \frac{1 + 19}{6}, sin (θ) = \frac{19 - 1}{6}

sin (θ) = - \frac{1 + 19}{6} : θ = arcsin (- \frac{1 + 19}{6}) + 2 πn, θ = π + arcsin (\frac{1 + 19}{6}) + 2 πn

sin (θ) = \frac{19 - 1}{6} : θ = arcsin (\frac{19 - 1}{6}) + 2 πn, θ = π - arcsin (\frac{19 - 1}{6}) + 2 πn

Combine all the solutions

θ = arcsin (- \frac{1 + 19}{6}) + 2 πn, θ = π + arcsin (\frac{1 + 19}{6}) + 2 πn, θ = arcsin (\frac{19 - 1}{6}) + 2 πn, θ = π - arcsin (\frac{19 - 1}{6}) + 2 πn

Show solutions in decimal form

θ = - 1.10430 \dots + 2 πn, θ = π + 1.10430 \dots + 2 πn, θ = 0.59416 \dots + 2 πn, θ = π - 0.59416 \dots + 2 πn

sin^{2} (x) - 4 = 0

tan (a) = \frac{2.8}{9}

cos (2 x) - 9 cos (x) + 5 = 0

u, x = 5 tanh (u)

2 sin (x) - csc (x) = 2