Trigonometry Examples

$\sin (\frac{θ}{2}) = 0$

Step 1

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

$\frac{θ}{2} = \arcsin (0)$

Step 2

Simplify the right side.

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Step 2.1

The exact value of $\arcsin (0)$ is $0$ .

$\frac{θ}{2} = 0$

Step 3

Set the numerator equal to zero.

$θ = 0$

Step 4

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $180$ to find the solution in the second quadrant.

$\frac{θ}{2} = 180 - 0$

Step 5

Solve for $θ$ .

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Step 5.1

Multiply both sides of the equation by $2$ .

$2 \frac{θ}{2} = 2 (180 - 0)$

Step 5.2

Simplify both sides of the equation.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Cancel the common factor of $2$ .

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Step 5.2.1.1.1

Cancel the common factor.

$2 \frac{θ}{2} = 2 (180 - 0)$

Step 5.2.1.1.2

Rewrite the expression.

$θ = 2 (180 - 0)$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify $2 (180 - 0)$ .

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Step 5.2.2.1.1

Subtract $0$ from $180$ .

$θ = 2 \cdot 180$

Step 5.2.2.1.2

Multiply $2$ by $180$ .

$θ = 360$

Step 6

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

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Step 6.1

The period of the function can be calculated using $\frac{360}{| b |}$ .

$\frac{360}{| b |}$

Step 6.2

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

$\frac{360}{| \frac{1}{2} |}$

Step 6.3

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

$\frac{360}{\frac{1}{2}}$

Step 6.4

Multiply the numerator by the reciprocal of the denominator.

$360 \cdot 2$

Step 6.5

Multiply $360$ by $2$ .

$720$

Step 7

The period of the $\sin (\frac{θ}{2})$ function is $720$ so values will repeat every $720$ degrees in both directions.

$θ = 720 n, 360 + 720 n$ , for any integer $n$

Step 8

Consolidate the answers.

$θ = 360 n$ , for any integer $n$