Trigonometry Examples

$\frac{\sin (53.1301)}{10} = \frac{\sin (x)}{8}$

Step 1

Rewrite the equation as $\frac{\sin (x)}{8} = \frac{\sin (53.1301)}{10}$ .

$\frac{\sin (x)}{8} = \frac{\sin (53.1301)}{10}$

Step 2

Multiply both sides of the equation by $8$ .

$8 \frac{\sin (x)}{8} = 8 \frac{\sin (53.1301)}{10}$

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 $8$ .

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Step 3.1.1.1

Cancel the common factor.

$8 \frac{\sin (x)}{8} = 8 \frac{\sin (53.1301)}{10}$

Step 3.1.1.2

Rewrite the expression.

$\sin (x) = 8 \frac{\sin (53.1301)}{10}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $8 \frac{\sin (53.1301)}{10}$ .

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Step 3.2.1.1

Cancel the common factor of $2$ .

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Step 3.2.1.1.1

Factor $2$ out of $8$ .

$\sin (x) = 2 (4) \frac{\sin (53.1301)}{10}$

Step 3.2.1.1.2

Factor $2$ out of $10$ .

$\sin (x) = 2 \cdot 4 \frac{\sin (53.1301)}{2 \cdot 5}$

Step 3.2.1.1.3

Cancel the common factor.

$\sin (x) = 2 \cdot 4 \frac{\sin (53.1301)}{2 \cdot 5}$

Step 3.2.1.1.4

Rewrite the expression.

$\sin (x) = 4 \frac{\sin (53.1301)}{5}$

Step 3.2.1.2

Combine $4$ and $\frac{\sin (53.1301)}{5}$ .

$\sin (x) = \frac{4 \sin (53.1301)}{5}$

Step 3.2.1.3

Evaluate $\sin (53.1301)$ .

$\sin (x) = \frac{4 \cdot 0.79999997}{5}$

Step 3.2.1.4

Multiply $4$ by $0.79999997$ .

$\sin (x) = \frac{3.1999999}{5}$

Step 3.2.1.5

Divide $3.1999999$ by $5$ .

$\sin (x) = 0.63999998$

Step 4

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

$x = \arcsin (0.63999998)$

Step 5

Simplify the right side.

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Step 5.1

Evaluate $\arcsin (0.63999998)$ .

$x = 39.79181802$

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.

$x = 180 - 39.79181802$

Step 7

Subtract $39.79181802$ from $180$ .

$x = 140.20818197$

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.

$x = 39.79181802 + 360 n, 140.20818197 + 360 n$ , for any integer $n$