Take the inverse sine of both sides of the equation to extract <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> from inside the sine.

The exact value of <math><mstyle displaystyle="true"><mi>arcsin</mi><mrow><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mfrac><mrow><mi>π</mi></mrow><mrow><mn>6</mn></mrow></mfrac></mstyle></math> .

Multiply both sides of the equation by <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Cancel the common factor of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>6</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from <math><mstyle displaystyle="true"><mi>π</mi></mstyle></math> to find the solution in the second quadrant.

Multiply both sides of the equation by <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Simplify both sides of the equation.

Cancel the common factor of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Simplify <math><mstyle displaystyle="true"><mn>2</mn><mo>⋅</mo><mrow><mo>(</mo><mi>π</mi><mo>-</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>6</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math> .

To write <math><mstyle displaystyle="true"><mfrac><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></mfrac></mstyle></math> as a fraction with a common denominator, multiply by <math><mstyle displaystyle="true"><mfrac><mrow><mn>6</mn></mrow><mrow><mn>6</mn></mrow></mfrac></mstyle></math> .

Write each expression with a common denominator of <math><mstyle displaystyle="true"><mn>6</mn></mstyle></math> , by multiplying each by an appropriate factor of <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Combine.

Multiply <math><mstyle displaystyle="true"><mn>6</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Combine the numerators over the common denominator.

Simplify the numerator.

Move <math><mstyle displaystyle="true"><mn>6</mn></mstyle></math> to the left of <math><mstyle displaystyle="true"><mi>π</mi></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mi>π</mi></mstyle></math> from <math><mstyle displaystyle="true"><mn>6</mn><mi>π</mi></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>6</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

The period of the function can be calculated using <math><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mrow><mrow><mo>|</mo><mi>b</mi><mo>|</mo></mrow></mrow></mfrac></mstyle></math> .

Replace <math><mstyle displaystyle="true"><mi>b</mi></mstyle></math> with <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> in the formula for period.

Solve the equation.

Multiply the numerator by the reciprocal of the denominator.

Multiply <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

The period of the <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mfrac><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math> function is <math><mstyle displaystyle="true"><mn>4</mn><mi>π</mi></mstyle></math> so values will repeat every <math><mstyle displaystyle="true"><mn>4</mn><mi>π</mi></mstyle></math> radians in both directions.

Do you know how to Solve for ? sin(x/2)=1/2? If not, you can write to our math experts in our application. The best solution for your task you can find above on this page.