Trigonometry Examples

$\frac{5}{\sin (B)} = \frac{81.3}{\sin (96)}$

Step 1

Factor each term.

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Step 1.1

Separate fractions.

$\frac{5}{\sin (B)} = \frac{81.3}{1} \cdot \frac{1}{\sin (96)}$

Step 1.2

Convert from $\frac{1}{\sin (96)}$ to $\csc (96)$ .

$\frac{5}{\sin (B)} = \frac{81.3}{1} \csc (96)$

Step 1.3

Divide $81.3$ by $1$ .

$\frac{5}{\sin (B)} = 81.3 \csc (96)$

Step 1.4

Evaluate $\csc (96)$ .

$\frac{5}{\sin (B)} = 81.3 \cdot 1.00550827$

Step 1.5

Multiply $81.3$ by $1.00550827$ .

$\frac{5}{\sin (B)} = 81.74782312$

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.

$\sin (B), 1$

Step 2.2

The LCM of one and any expression is the expression.

$\sin (B)$

Step 3

Multiply each term in $\frac{5}{\sin (B)} = 81.74782312$ by $\sin (B)$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{5}{\sin (B)} = 81.74782312$ by $\sin (B)$ .

$\frac{5}{\sin (B)} \sin (B) = 81.74782312 \sin (B)$

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

Cancel the common factor.

$\frac{5}{\sin (B)} \sin (B) = 81.74782312 \sin (B)$

Step 3.2.1.2

Rewrite the expression.

$5 = 81.74782312 \sin (B)$

Step 4

Solve the equation.

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Step 4.1

Rewrite the equation as $81.74782312 \sin (B) = 5$ .

$81.74782312 \sin (B) = 5$

Step 4.2

Divide each term in $81.74782312 \sin (B) = 5$ by $81.74782312$ and simplify.

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Step 4.2.1

Divide each term in $81.74782312 \sin (B) = 5$ by $81.74782312$ .

$\frac{81.74782312 \sin (B)}{81.74782312} = \frac{5}{81.74782312}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $81.74782312$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{81.74782312 \sin (B)}{81.74782312} = \frac{5}{81.74782312}$

Step 4.2.2.1.2

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

$\sin (B) = \frac{5}{81.74782312}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $5$ by $81.74782312$ .

$\sin (B) = 0.0611637$

Step 5

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

$B = \arcsin (0.0611637)$

Step 6

Simplify the right side.

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Step 6.1

Evaluate $\arcsin (0.0611637)$ .

$B = 3.50661103$

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.

$B = 180 - 3.50661103$

Step 8

Subtract $3.50661103$ from $180$ .

$B = 176.49338896$

Step 9

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

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

$B = 3.50661103 + 360 n, 176.49338896 + 360 n$ , for any integer $n$