Trigonometry Examples

$\frac{15.2}{\sin (H)} = \frac{12}{\sin (50 °)}$

Step 1

Factor each term.

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Step 1.1

Separate fractions.

$\frac{15.2}{\sin (H)} = \frac{12}{1} \cdot \frac{1}{\sin (50 °)}$

Step 1.2

Convert from $\frac{1}{\sin (50 °)}$ to $\csc (50 °)$ .

$\frac{15.2}{\sin (H)} = \frac{12}{1} \csc (50 °)$

Step 1.3

Divide $12$ by $1$ .

$\frac{15.2}{\sin (H)} = 12 \csc (50 °)$

Step 1.4

Evaluate $\csc (50 °)$ .

$\frac{15.2}{\sin (H)} = 12 \cdot 1.30540728$

Step 1.5

Multiply $12$ by $1.30540728$ .

$\frac{15.2}{\sin (H)} = 15.66488747$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$\sin (H), 1$

Step 2.2

The LCM of one and any expression is the expression.

$\sin (H)$

Step 3

Multiply each term in $\frac{15.2}{\sin (H)} = 15.66488747$ by $\sin (H)$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{15.2}{\sin (H)} = 15.66488747$ by $\sin (H)$ .

$\frac{15.2}{\sin (H)} \sin (H) = 15.66488747 \sin (H)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $\sin (H)$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{15.2}{\sin (H)} \sin (H) = 15.66488747 \sin (H)$

Step 3.2.1.2

Rewrite the expression.

$15.2 = 15.66488747 \sin (H)$

Step 4

Solve the equation.

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Step 4.1

Rewrite the equation as $15.66488747 \sin (H) = 15.2$ .

$15.66488747 \sin (H) = 15.2$

Step 4.2

Divide each term in $15.66488747 \sin (H) = 15.2$ by $15.66488747$ and simplify.

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Step 4.2.1

Divide each term in $15.66488747 \sin (H) = 15.2$ by $15.66488747$ .

$\frac{15.66488747 \sin (H)}{15.66488747} = \frac{15.2}{15.66488747}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $15.66488747$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{15.66488747 \sin (H)}{15.66488747} = \frac{15.2}{15.66488747}$

Step 4.2.2.1.2

Divide $\sin (H)$ by $1$ .

$\sin (H) = \frac{15.2}{15.66488747}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $15.2$ by $15.66488747$ .

$\sin (H) = 0.97032296$

Step 5

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

$H = \arcsin (0.97032296)$

Step 6

Simplify the right side.

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Step 6.1

Evaluate $\arcsin (0.97032296)$ .

$H = 76.00645168$

Step 7

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.

$H = 180 - 76.00645168$

Step 8

Subtract $76.00645168$ from $180$ .

$H = 103.99354831$

Step 9

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

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

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

$\frac{360}{1}$

Step 9.4

Divide $360$ by $1$ .

$360$

Step 10

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

$H = 76.00645168 + 360 n, 103.99354831 + 360 n$ , for any integer $n$