Trigonometry Examples

$\frac{\sin (30)}{8} = \frac{\sin (C)}{12}$

Step 1

Rewrite the equation as $\frac{\sin (C)}{12} = \frac{\sin (30)}{8}$ .

$\frac{\sin (C)}{12} = \frac{\sin (30)}{8}$

Step 2

Multiply both sides of the equation by $12$ .

$12 \frac{\sin (C)}{12} = 12 \frac{\sin (30)}{8}$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $12$ .

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Step 3.1.1.1

Cancel the common factor.

$12 \frac{\sin (C)}{12} = 12 \frac{\sin (30)}{8}$

Step 3.1.1.2

Rewrite the expression.

$\sin (C) = 12 \frac{\sin (30)}{8}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $12 \frac{\sin (30)}{8}$ .

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Step 3.2.1.1

Cancel the common factor of $4$ .

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Step 3.2.1.1.1

Factor $4$ out of $12$ .

$\sin (C) = 4 (3) \frac{\sin (30)}{8}$

Step 3.2.1.1.2

Factor $4$ out of $8$ .

$\sin (C) = 4 \cdot 3 \frac{\sin (30)}{4 \cdot 2}$

Step 3.2.1.1.3

Cancel the common factor.

$\sin (C) = 4 \cdot 3 \frac{\sin (30)}{4 \cdot 2}$

Step 3.2.1.1.4

Rewrite the expression.

$\sin (C) = 3 \frac{\sin (30)}{2}$

Step 3.2.1.2

Combine $3$ and $\frac{\sin (30)}{2}$ .

$\sin (C) = \frac{3 \sin (30)}{2}$

Step 3.2.1.3

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$\sin (C) = \frac{3 (\frac{1}{2})}{2}$

Step 3.2.1.4

Combine $3$ and $\frac{1}{2}$ .

$\sin (C) = \frac{\frac{3}{2}}{2}$

Step 3.2.1.5

Multiply the numerator by the reciprocal of the denominator.

$\sin (C) = \frac{3}{2} \cdot \frac{1}{2}$

Step 3.2.1.6

Multiply $\frac{3}{2} \cdot \frac{1}{2}$ .

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Step 3.2.1.6.1

Multiply $\frac{3}{2}$ by $\frac{1}{2}$ .

$\sin (C) = \frac{3}{2 \cdot 2}$

Step 3.2.1.6.2

Multiply $2$ by $2$ .

$\sin (C) = \frac{3}{4}$

Step 4

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

$C = \arcsin (\frac{3}{4})$

Step 5

Simplify the right side.

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Step 5.1

Evaluate $\arcsin (\frac{3}{4})$ .

$C = 48.59037789$

Step 6

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.

$C = 180 - 48.59037789$

Step 7

Subtract $48.59037789$ from $180$ .

$C = 131.4096221$

Step 8

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

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Step 8.1

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

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

Step 8.2

Replace $b$ with $1$ in the formula for period.

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

Step 8.3

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

$\frac{360}{1}$

Step 8.4

Divide $360$ by $1$ .

$360$

Step 9

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

$C = 48.59037789 + 360 n, 131.4096221 + 360 n$ , for any integer $n$