Trigonometry Examples

$\sin (r) = \frac{123 \sin (60)}{184.5}$

Step 1

Simplify $\frac{123 \sin (60)}{184.5}$ .

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Step 1.1

The exact value of $\sin (60)$ is $\frac{\sqrt{3}}{2}$ .

$\sin (r) = \frac{123 \frac{\sqrt{3}}{2}}{184.5}$

Step 1.2

Combine $123$ and $\frac{\sqrt{3}}{2}$ .

$\sin (r) = \frac{\frac{123 \sqrt{3}}{2}}{184.5}$

Step 1.3

Multiply the numerator by the reciprocal of the denominator.

$\sin (r) = \frac{123 \sqrt{3}}{2} \cdot \frac{1}{184.5}$

Step 1.4

Divide $1$ by $184.5$ .

$\sin (r) = \frac{123 \sqrt{3}}{2} \cdot 0.\overline{00542}$

Step 1.5

Multiply $\frac{123 \sqrt{3}}{2} \cdot 0.\overline{00542}$ .

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Step 1.5.1

Combine $\frac{123 \sqrt{3}}{2}$ and $0.\overline{00542}$ .

$\sin (r) = \frac{123 \sqrt{3} \cdot 0.\overline{00542}}{2}$

Step 1.5.2

Multiply $0.\overline{00542}$ by $123$ .

$\sin (r) = \frac{0.66666666 \sqrt{3}}{2}$

Step 1.5.3

Multiply $0.66666666$ by $\sqrt{3}$ .

$\sin (r) = \frac{1.15470053}{2}$

Step 1.6

Divide $1.15470053$ by $2$ .

$\sin (r) = 0.57735026$

Step 2

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

$r = \arcsin (0.57735026)$

Step 3

Simplify the right side.

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Step 3.1

Evaluate $\arcsin (0.57735026)$ .

$r = 35.26438968$

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.

$r = 180 - 35.26438968$

Step 5

Subtract $35.26438968$ from $180$ .

$r = 144.73561031$

Step 6

Find the period of $\sin (r)$ .

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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 $1$ in the formula for period.

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

Step 6.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{360}{1}$

Step 6.4

Divide $360$ by $1$ .

$360$

Step 7

The period of the $\sin (r)$ function is $360$ so values will repeat every $360$ degrees in both directions.

$r = 35.26438968 + 360 n, 144.73561031 + 360 n$ , for any integer $n$