What is the general solution for sin^3(x)+sin(x)=2cos^2(x) ?

The general solution for sin^3(x)+sin(x)=2cos^2(x) is x=0.76928…+2pin,x=pi-0.76928…+2pin

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sin^3(x)+sin(x)=2cos^2(x)

sin^{3} (x) + sin (x) = 2 cos^{2} (x)

x = 0.76928 \dots + 2 πn, x = π - 0.76928 \dots + 2 πn

Solution steps

sin^{3} (x) + sin (x) = 2 cos^{2} (x)

Subtract

2 cos^{2} (x)

from both sides

sin^{3} (x) + sin (x) - 2 cos^{2} (x) = 0

Rewrite using trig identities

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

Solve by substitution

sin (x) \approx 0.69562 \dots

sin (x) = 0.69562 \dots : x = arcsin (0.69562 \dots) + 2 πn, x = π - arcsin (0.69562 \dots) + 2 πn

Combine all the solutions

x = arcsin (0.69562 \dots) + 2 πn, x = π - arcsin (0.69562 \dots) + 2 πn

Show solutions in decimal form

x = 0.76928 \dots + 2 πn, x = π - 0.76928 \dots + 2 πn

4 sin^{2} (x) + 2 cos^{2} (x) = 3

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

tan^{2} (x) cos (x) + cos^{2} (x) = 1

sinh (x) = \frac{3}{2}

a = 2 sin (3 0^{\circ} + x) - cos (x)

What is the general solution for sin^3(x)+sin(x)=2cos^2(x) ?
The general solution for sin^3(x)+sin(x)=2cos^2(x) is x=0.76928…+2pin,x=pi-0.76928…+2pin