What is the general solution for 2sin^3(x)+cos^2(x)=1 ?

The general solution for 2sin^3(x)+cos^2(x)=1 is x=2pin,x=pi+2pin,x= pi/6+2pin,x=(5pi)/6+2pin

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2sin^3(x)+cos^2(x)=1

2 sin^{3} (x) + cos^{2} (x) = 1

x = 2 πn, x = π + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Solution steps

2 sin^{3} (x) + cos^{2} (x) = 1

Subtract

1

from both sides

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

Rewrite using trig identities

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

Solve by substitution

sin (x) = 0, sin (x) = \frac{1}{2}

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = \frac{1}{2} : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

2 tan (x) = tan (x) tan (x)

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

tan (x) = \frac{59}{56}

cos (θ) = \frac{5}{5 2}

arccos (x) = arcsin (\frac{9}{41})

What is the general solution for 2sin^3(x)+cos^2(x)=1 ?
The general solution for 2sin^3(x)+cos^2(x)=1 is x=2pin,x=pi+2pin,x= pi/6+2pin,x=(5pi)/6+2pin