What is the general solution for 3sin^2(x)+2sin(x)cos^2(x/2)-sin(x)=0 ?

The general solution for 3sin^2(x)+2sin(x)cos^2(x/2)-sin(x)=0 is x=2pin,x=pi+2pin,x=-0.32175…+pin

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

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

x = 2 πn, x = π + 2 πn, x = - 0.32175 \dots + πn

Solution steps

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

Factor

3 sin^{2} (x) + 2 sin (x) cos^{2} (\frac{x}{2}) - sin (x) : sin (x) (3 sin (x) + 2 cos^{2} (\frac{x}{2}) - 1)

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

Solving each part separately

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

sin (x) = 0 : x = 2 πn, x = π + 2 πn

3 sin (x) + 2 cos^{2} (\frac{x}{2}) - 1 = 0 : x = arctan (- \frac{1}{3}) + πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = arctan (- \frac{1}{3}) + πn

Show solutions in decimal form

x = 2 πn, x = π + 2 πn, x = - 0.32175 \dots + πn

3 tan^{2} (y) = 5 sec (y) - 1

\frac{( 1 - tan ^{2} ( a ) )}{( tan ( a ) )} = 2 cot^{2} (a)

cos (6 x) = cos (2 x)

cot (x) cos (x) - sin (x) = 1

cos^{4} (x) + cos^{3} (x) - 2 = 0

What is the general solution for 3sin^2(x)+2sin(x)cos^2(x/2)-sin(x)=0 ?
The general solution for 3sin^2(x)+2sin(x)cos^2(x/2)-sin(x)=0 is x=2pin,x=pi+2pin,x=-0.32175…+pin