Find All Complex Solutions sin(x/2) = square root of 3-sin(x/2)

$\sin (\frac{x}{2}) = \sqrt{3} - \sin (\frac{x}{2})$

Move all terms containing $\sin (\frac{x}{2})$ to the left side of the equation.

Add $\sin (\frac{x}{2})$ to both sides of the equation.

$\sin (\frac{x}{2}) + \sin (\frac{x}{2}) = \sqrt{3}$

Add $\sin (\frac{x}{2})$ and $\sin (\frac{x}{2})$ .

$2 \sin (\frac{x}{2}) = \sqrt{3}$

Divide each term in $2 \sin (\frac{x}{2}) = \sqrt{3}$ by $2$ and simplify.

Divide each term in $2 \sin (\frac{x}{2}) = \sqrt{3}$ by $2$ .

$\frac{2 \sin (\frac{x}{2})}{2} = \frac{\sqrt{3}}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 \sin (\frac{x}{2})}{2} = \frac{\sqrt{3}}{2}$

Divide $\sin (\frac{x}{2})$ by $1$ .

$\sin (\frac{x}{2}) = \frac{\sqrt{3}}{2}$

Take the inverse sine of both sides of the equation to extract $x$ from inside the sine.

$\frac{x}{2} = \arcsin (\frac{\sqrt{3}}{2})$

Simplify the right side.

The exact value of $\arcsin (\frac{\sqrt{3}}{2})$ is $\frac{π}{3}$ .

$\frac{x}{2} = \frac{π}{3}$

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 \frac{π}{3}$

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$2 \frac{x}{2} = 2 \frac{π}{3}$

Rewrite the expression.

$x = 2 \frac{π}{3}$

Simplify the right side.

Combine $2$ and $\frac{π}{3}$ .

$x = \frac{2 π}{3}$

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$\frac{x}{2} = π - \frac{π}{3}$

Solve for $x$ .

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 (π - \frac{π}{3})$

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$2 \frac{x}{2} = 2 (π - \frac{π}{3})$

Rewrite the expression.

$x = 2 (π - \frac{π}{3})$

Simplify the right side.

Simplify $2 (π - \frac{π}{3})$ .

To write $π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = 2 (π \cdot \frac{3}{3} - \frac{π}{3})$

Combine fractions.

Combine $π$ and $\frac{3}{3}$ .

$x = 2 (\frac{π \cdot 3}{3} - \frac{π}{3})$

Combine the numerators over the common denominator.

$x = 2 \frac{π \cdot 3 - π}{3}$

Simplify the numerator.

Move $3$ to the left of $π$ .

$x = 2 \frac{3 \cdot π - π}{3}$

Subtract $π$ from $3 π$ .

$x = 2 \frac{2 π}{3}$

Multiply $2 \frac{2 π}{3}$ .

Combine $2$ and $\frac{2 π}{3}$ .

$x = \frac{2 (2 π)}{3}$

Multiply $2$ by $2$ .

$x = \frac{4 π}{3}$

Find the period of $\sin (\frac{x}{2})$ .

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{1}{2}$ in the formula for period.

$\frac{2 π}{| \frac{1}{2} |}$

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Multiply $2$ by $2$ .

$4 π$

The period of the $\sin (\frac{x}{2})$ function is $4 π$ so values will repeat every $4 π$ radians in both directions.

$x = \frac{2 π}{3} + 4 π n, \frac{4 π}{3} + 4 π n$ , for any integer $n$

Find All Complex Solutions sin(x/2) = square root of 3-sin(x/2)

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