What is the general solution for (sin(θ)+cos^2(θ))^2=2sin(θ) ?

The general solution for (sin(θ)+cos^2(θ))^2=2sin(θ) is θ=0.80561…+2pin,θ=pi-0.80561…+2pin

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(sin(θ)+cos^2(θ))^2=2sin(θ)

(sin (θ) + cos^{2} (θ))^{2} = 2 sin (θ)

θ = 0.80561 \dots + 2 πn, θ = π - 0.80561 \dots + 2 πn

Solution steps

(sin (θ) + cos^{2} (θ))^{2} = 2 sin (θ)

Subtract

2 sin (θ)

from both sides

(sin (θ) + cos^{2} (θ))^{2} - 2 sin (θ) = 0

Rewrite using trig identities

1 - sin^{2} (θ) + sin^{4} (θ) - 2 sin^{3} (θ) = 0

Solve by substitution

sin (θ) \approx 0.72125 \dots, sin (θ) \approx 2.34859 \dots

sin (θ) = 0.72125 \dots : θ = arcsin (0.72125 \dots) + 2 πn, θ = π - arcsin (0.72125 \dots) + 2 πn

sin (θ) = 2.34859 \dots :

No Solution

Combine all the solutions

θ = arcsin (0.72125 \dots) + 2 πn, θ = π - arcsin (0.72125 \dots) + 2 πn

Show solutions in decimal form

θ = 0.80561 \dots + 2 πn, θ = π - 0.80561 \dots + 2 πn

\frac{2}{3} = sin (2 θ)

sin (3 x) - 2 = - 3 sin (3 x)

sin (a) = \frac{1}{5}

θ, cos (θ) = \frac{2}{3}

x, arcsin (x) = arccos (y)

What is the general solution for (sin(θ)+cos^2(θ))^2=2sin(θ) ?
The general solution for (sin(θ)+cos^2(θ))^2=2sin(θ) is θ=0.80561…+2pin,θ=pi-0.80561…+2pin