Trigonometry Examples

$\frac{\sin (112)}{32} = \frac{\sin (x)}{17}$

Step 1

Rewrite the equation as $\frac{\sin (x)}{17} = \frac{\sin (112)}{32}$ .

$\frac{\sin (x)}{17} = \frac{\sin (112)}{32}$

Step 2

Multiply both sides of the equation by $17$ .

$17 \frac{\sin (x)}{17} = 17 \frac{\sin (112)}{32}$

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 $17$ .

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Step 3.1.1.1

Cancel the common factor.

$17 \frac{\sin (x)}{17} = 17 \frac{\sin (112)}{32}$

Step 3.1.1.2

Rewrite the expression.

$\sin (x) = 17 \frac{\sin (112)}{32}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $17 \frac{\sin (112)}{32}$ .

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Step 3.2.1.1

Evaluate $\sin (112)$ .

$\sin (x) = 17 (\frac{0.92718385}{32})$

Step 3.2.1.2

Divide $0.92718385$ by $32$ .

$\sin (x) = 17 \cdot 0.02897449$

Step 3.2.1.3

Multiply $17$ by $0.02897449$ .

$\sin (x) = 0.49256642$

Step 4

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

$x = \arcsin (0.49256642)$

Step 5

Simplify the right side.

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Step 5.1

Evaluate $\arcsin (0.49256642)$ .

$x = 29.50940512$

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.

$x = 180 - 29.50940512$

Step 7

Subtract $29.50940512$ from $180$ .

$x = 150.49059487$

Step 8

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

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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 $\sin (x)$ function is $360$ so values will repeat every $360$ degrees in both directions.

$x = 29.50940512 + 360 n, 150.49059487 + 360 n$ , for any integer $n$