Trigonometry Examples

$\frac{\sin (150)}{270} = \frac{\sin (y)}{150}$

Step 1

Rewrite the equation as $\frac{\sin (y)}{150} = \frac{\sin (150)}{270}$ .

$\frac{\sin (y)}{150} = \frac{\sin (150)}{270}$

Step 2

Multiply both sides of the equation by $150$ .

$150 \frac{\sin (y)}{150} = 150 \frac{\sin (150)}{270}$

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 $150$ .

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Step 3.1.1.1

Cancel the common factor.

$150 \frac{\sin (y)}{150} = 150 \frac{\sin (150)}{270}$

Step 3.1.1.2

Rewrite the expression.

$\sin (y) = 150 \frac{\sin (150)}{270}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $150 \frac{\sin (150)}{270}$ .

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Step 3.2.1.1

Simplify terms.

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Step 3.2.1.1.1

Cancel the common factor of $30$ .

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Step 3.2.1.1.1.1

Factor $30$ out of $150$ .

$\sin (y) = 30 (5) \frac{\sin (150)}{270}$

Step 3.2.1.1.1.2

Factor $30$ out of $270$ .

$\sin (y) = 30 \cdot 5 \frac{\sin (150)}{30 \cdot 9}$

Step 3.2.1.1.1.3

Cancel the common factor.

$\sin (y) = 30 \cdot 5 \frac{\sin (150)}{30 \cdot 9}$

Step 3.2.1.1.1.4

Rewrite the expression.

$\sin (y) = 5 \frac{\sin (150)}{9}$

Step 3.2.1.1.2

Combine $5$ and $\frac{\sin (150)}{9}$ .

$\sin (y) = \frac{5 \sin (150)}{9}$

Step 3.2.1.2

Simplify the numerator.

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Step 3.2.1.2.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\sin (y) = \frac{5 \sin (30)}{9}$

Step 3.2.1.2.2

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

$\sin (y) = \frac{5 (\frac{1}{2})}{9}$

Step 3.2.1.3

Combine $5$ and $\frac{1}{2}$ .

$\sin (y) = \frac{\frac{5}{2}}{9}$

Step 3.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$\sin (y) = \frac{5}{2} \cdot \frac{1}{9}$

Step 3.2.1.5

Multiply $\frac{5}{2} \cdot \frac{1}{9}$ .

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Step 3.2.1.5.1

Multiply $\frac{5}{2}$ by $\frac{1}{9}$ .

$\sin (y) = \frac{5}{2 \cdot 9}$

Step 3.2.1.5.2

Multiply $2$ by $9$ .

$\sin (y) = \frac{5}{18}$

Step 4

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

$y = \arcsin (\frac{5}{18})$

Step 5

Simplify the right side.

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Step 5.1

Evaluate $\arcsin (\frac{5}{18})$ .

$y = 16.12762021$

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.

$y = 180 - 16.12762021$

Step 7

Subtract $16.12762021$ from $180$ .

$y = 163.87237978$

Step 8

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

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

$y = 16.12762021 + 360 n, 163.87237978 + 360 n$ , for any integer $n$