Trigonometry Examples

$\frac{8.59}{\sin (48)} = \frac{10}{\sin (b)}$

Step 1

Rewrite the equation as $\frac{10}{\sin (b)} = \frac{8.59}{\sin (48)}$ .

$\frac{10}{\sin (b)} = \frac{8.59}{\sin (48)}$

Step 2

Factor each term.

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Step 2.1

Separate fractions.

$\frac{10}{\sin (b)} = \frac{8.59}{1} \cdot \frac{1}{\sin (48)}$

Step 2.2

Convert from $\frac{1}{\sin (48)}$ to $\csc (48)$ .

$\frac{10}{\sin (b)} = \frac{8.59}{1} \csc (48)$

Step 2.3

Divide $8.59$ by $1$ .

$\frac{10}{\sin (b)} = 8.59 \csc (48)$

Step 2.4

Evaluate $\csc (48)$ .

$\frac{10}{\sin (b)} = 8.59 \cdot 1.34563272$

Step 2.5

Multiply $8.59$ by $1.34563272$ .

$\frac{10}{\sin (b)} = 11.55898514$

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 $\frac{10}{\sin (b)} = 11.55898514$ by $\sin (b)$ to eliminate the fractions.

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Step 4.1

Multiply each term in $\frac{10}{\sin (b)} = 11.55898514$ by $\sin (b)$ .

$\frac{10}{\sin (b)} \sin (b) = 11.55898514 \sin (b)$

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.

$\frac{10}{\sin (b)} \sin (b) = 11.55898514 \sin (b)$

Step 4.2.1.2

Rewrite the expression.

$10 = 11.55898514 \sin (b)$

Step 5

Solve the equation.

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Step 5.1

Rewrite the equation as $11.55898514 \sin (b) = 10$ .

$11.55898514 \sin (b) = 10$

Step 5.2

Divide each term in $11.55898514 \sin (b) = 10$ by $11.55898514$ and simplify.

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Step 5.2.1

Divide each term in $11.55898514 \sin (b) = 10$ by $11.55898514$ .

$\frac{11.55898514 \sin (b)}{11.55898514} = \frac{10}{11.55898514}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $11.55898514$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{11.55898514 \sin (b)}{11.55898514} = \frac{10}{11.55898514}$

Step 5.2.2.1.2

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

$\sin (b) = \frac{10}{11.55898514}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $10$ by $11.55898514$ .

$\sin (b) = 0.86512785$

Step 6

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

$b = \arcsin (0.86512785)$

Step 7

Simplify the right side.

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Step 7.1

Evaluate $\arcsin (0.86512785)$ .

$b = 59.89730756$

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.

$b = 180 - 59.89730756$

Step 9

Subtract $59.89730756$ from $180$ .

$b = 120.10269243$

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.

$b = 59.89730756 + 360 n, 120.10269243 + 360 n$ , for any integer $n$