What is the general solution for 4sin(2θ)-5sin(θ)=0 ?

The general solution for 4sin(2θ)-5sin(θ)=0 is θ=2pin,θ=pi+2pin,θ=0.89566…+2pin,θ=2pi-0.89566…+2pin

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4sin(2θ)-5sin(θ)=0

4 sin (2 θ) - 5 sin (θ) = 0

θ = 2 πn, θ = π + 2 πn, θ = 0.89566 \dots + 2 πn, θ = 2 π - 0.89566 \dots + 2 πn

Solution steps

4 sin (2 θ) - 5 sin (θ) = 0

Rewrite using trig identities

- 5 sin (θ) + 8 cos (θ) sin (θ) = 0

Factor

- 5 sin (θ) + 8 cos (θ) sin (θ) : sin (θ) (8 cos (θ) - 5)

sin (θ) (8 cos (θ) - 5) = 0

Solving each part separately

sin (θ) = 0 or 8 cos (θ) - 5 = 0

sin (θ) = 0 : θ = 2 πn, θ = π + 2 πn

8 cos (θ) - 5 = 0 : θ = arccos (\frac{5}{8}) + 2 πn, θ = 2 π - arccos (\frac{5}{8}) + 2 πn

Combine all the solutions

θ = 2 πn, θ = π + 2 πn, θ = arccos (\frac{5}{8}) + 2 πn, θ = 2 π - arccos (\frac{5}{8}) + 2 πn

Show solutions in decimal form

θ = 2 πn, θ = π + 2 πn, θ = 0.89566 \dots + 2 πn, θ = 2 π - 0.89566 \dots + 2 πn

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

tan (\frac{2 x}{3}) - 1 = 0

sin (θ) - cos (θ) = 1

- \frac{5}{2} csc (\frac{θ}{2}) = - 3 - csc (\frac{θ}{2})

4 cos (2 θ) + 11 sin (θ) - 4 = 3

What is the general solution for 4sin(2θ)-5sin(θ)=0 ?
The general solution for 4sin(2θ)-5sin(θ)=0 is θ=2pin,θ=pi+2pin,θ=0.89566…+2pin,θ=2pi-0.89566…+2pin