Precalculus Examples

$\frac{4}{\sin (B)} = \frac{18}{\sin (69 °)}$

Step 1

Factor each term.

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Step 1.1

Separate fractions.

$\frac{4}{\sin (B)} = \frac{18}{1} \cdot \frac{1}{\sin (69 °)}$

Step 1.2

Convert from $\frac{1}{\sin (69 °)}$ to $\csc (69 °)$ .

$\frac{4}{\sin (B)} = \frac{18}{1} \csc (69 °)$

Step 1.3

Divide $18$ by $1$ .

$\frac{4}{\sin (B)} = 18 \csc (69 °)$

Step 1.4

Evaluate $\csc (69 °)$ .

$\frac{4}{\sin (B)} = 18 \cdot 1.07114499$

Step 1.5

Multiply $18$ by $1.07114499$ .

$\frac{4}{\sin (B)} = 19.28060988$

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 $\frac{4}{\sin (B)} = 19.28060988$ by $\sin (B)$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{4}{\sin (B)} = 19.28060988$ by $\sin (B)$ .

$\frac{4}{\sin (B)} \sin (B) = 19.28060988 \sin (B)$

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.

$\frac{4}{\sin (B)} \sin (B) = 19.28060988 \sin (B)$

Step 3.2.1.2

Rewrite the expression.

$4 = 19.28060988 \sin (B)$

Step 4

Solve the equation.

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Step 4.1

Rewrite the equation as $19.28060988 \sin (B) = 4$ .

$19.28060988 \sin (B) = 4$

Step 4.2

Divide each term in $19.28060988 \sin (B) = 4$ by $19.28060988$ and simplify.

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Step 4.2.1

Divide each term in $19.28060988 \sin (B) = 4$ by $19.28060988$ .

$\frac{19.28060988 \sin (B)}{19.28060988} = \frac{4}{19.28060988}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $19.28060988$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{19.28060988 \sin (B)}{19.28060988} = \frac{4}{19.28060988}$

Step 4.2.2.1.2

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

$\sin (B) = \frac{4}{19.28060988}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $4$ by $19.28060988$ .

$\sin (B) = 0.20746231$

Step 5

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

$B = \arcsin (0.20746231)$

Step 6

Simplify the right side.

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Step 6.1

Evaluate $\arcsin (0.20746231)$ .

$B = 11.97367885$

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.

$B = 180 - 11.97367885$

Step 8

Subtract $11.97367885$ from $180$ .

$B = 168.02632114$

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.

$B = 11.97367885 + 360 n, 168.02632114 + 360 n$ , for any integer $n$