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

The general solution for sin(x)+sin^2(x/2)= 1/2 is x=0.46364…+pin

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sin(x)+sin^2(x/2)= 1/2

sin (x) + sin^{2} (\frac{x}{2}) = \frac{1}{2}

x = 0.46364 \dots + πn

Solution steps

sin (x) + sin^{2} (\frac{x}{2}) = \frac{1}{2}

Subtract

\frac{1}{2}

from both sides

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

Simplify

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

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

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

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

Rewrite using trig identities

- 1 + 2 tan (x) = 0

Move

1

to the right side

2 tan (x) = 1

Divide both sides by

2

tan (x) = \frac{1}{2}

Apply trig inverse properties

x = arctan (\frac{1}{2}) + πn

Show solutions in decimal form

x = 0.46364 \dots + πn

sin^{5} (x) + sin^{3} (x) = 0

5 sin^{2} (x) cos (7 x) - cos (7 x) = 0

x, cos^{2} (x) \cdot z^{2} - 2 cos^{2} (x) \cdot z + 1 = 0

sin (a) + sin (12 0^{\circ} + a) + sin (12 0^{\circ} - a) = 0

3 sin (x) sin (x) = 5 cos (x) - 2

What is the general solution for sin(x)+sin^2(x/2)= 1/2 ?
The general solution for sin(x)+sin^2(x/2)= 1/2 is x=0.46364…+pin