Trigonometry Examples

$\frac{4}{\sin (10)} = \frac{3}{\sin (B)}$

Step 1

Rewrite the equation as $\frac{3}{\sin (B)} = \frac{4}{\sin (10)}$ .

$\frac{3}{\sin (B)} = \frac{4}{\sin (10)}$

Step 2

Factor each term.

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Step 2.1

Separate fractions.

$\frac{3}{\sin (B)} = \frac{4}{1} \cdot \frac{1}{\sin (10)}$

Step 2.2

Convert from $\frac{1}{\sin (10)}$ to $\csc (10)$ .

$\frac{3}{\sin (B)} = \frac{4}{1} \csc (10)$

Step 2.3

Divide $4$ by $1$ .

$\frac{3}{\sin (B)} = 4 \csc (10)$

Step 2.4

Evaluate $\csc (10)$ .

$\frac{3}{\sin (B)} = 4 \cdot 5.75877048$

Step 2.5

Multiply $4$ by $5.75877048$ .

$\frac{3}{\sin (B)} = 23.03508193$

Step 3

Find the LCD of the terms in the equation.

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Step 3.1

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 3.2

The LCM of one and any expression is the expression.

$\sin (B)$

Step 4

Multiply each term in $\frac{3}{\sin (B)} = 23.03508193$ by $\sin (B)$ to eliminate the fractions.

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Step 4.1

Multiply each term in $\frac{3}{\sin (B)} = 23.03508193$ by $\sin (B)$ .

$\frac{3}{\sin (B)} \sin (B) = 23.03508193 \sin (B)$

Step 4.2

Simplify the left side.

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Step 4.2.1

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

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Step 4.2.1.1

Cancel the common factor.

$\frac{3}{\sin (B)} \sin (B) = 23.03508193 \sin (B)$

Step 4.2.1.2

Rewrite the expression.

$3 = 23.03508193 \sin (B)$

Step 5

Solve the equation.

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Step 5.1

Rewrite the equation as $23.03508193 \sin (B) = 3$ .

$23.03508193 \sin (B) = 3$

Step 5.2

Divide each term in $23.03508193 \sin (B) = 3$ by $23.03508193$ and simplify.

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Step 5.2.1

Divide each term in $23.03508193 \sin (B) = 3$ by $23.03508193$ .

$\frac{23.03508193 \sin (B)}{23.03508193} = \frac{3}{23.03508193}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $23.03508193$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{23.03508193 \sin (B)}{23.03508193} = \frac{3}{23.03508193}$

Step 5.2.2.1.2

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

$\sin (B) = \frac{3}{23.03508193}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $3$ by $23.03508193$ .

$\sin (B) = 0.13023613$

Step 6

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

$B = \arcsin (0.13023613)$

Step 7

Simplify the right side.

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Step 7.1

Evaluate $\arcsin (0.13023613)$ .

$B = 7.48323776$

Step 8

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.

$B = 180 - 7.48323776$

Step 9

Subtract $7.48323776$ from $180$ .

$B = 172.51676223$

Step 10

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

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Step 10.1

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

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

Step 10.2

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

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

Step 10.3

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

$\frac{360}{1}$

Step 10.4

Divide $360$ by $1$ .

$360$

Step 11

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

$B = 7.48323776 + 360 n, 172.51676223 + 360 n$ , for any integer $n$