Trigonometry Examples

$\frac{10}{\sin (10)} = \frac{9}{\sin (b)}$

Step 1

Rewrite the equation as $\frac{9}{\sin (b)} = \frac{10}{\sin (10)}$ .

$\frac{9}{\sin (b)} = \frac{10}{\sin (10)}$

Step 2

Factor each term.

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Step 2.1

Separate fractions.

$\frac{9}{\sin (b)} = \frac{10}{1} \cdot \frac{1}{\sin (10)}$

Step 2.2

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

$\frac{9}{\sin (b)} = \frac{10}{1} \csc (10)$

Step 2.3

Divide $10$ by $1$ .

$\frac{9}{\sin (b)} = 10 \csc (10)$

Step 2.4

Evaluate $\csc (10)$ .

$\frac{9}{\sin (b)} = 10 \cdot 5.75877048$

Step 2.5

Multiply $10$ by $5.75877048$ .

$\frac{9}{\sin (b)} = 57.58770483$

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 $\frac{9}{\sin (b)} = 57.58770483$ by $\sin (b)$ to eliminate the fractions.

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Step 4.1

Multiply each term in $\frac{9}{\sin (b)} = 57.58770483$ by $\sin (b)$ .

$\frac{9}{\sin (b)} \sin (b) = 57.58770483 \sin (b)$

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.

$\frac{9}{\sin (b)} \sin (b) = 57.58770483 \sin (b)$

Step 4.2.1.2

Rewrite the expression.

$9 = 57.58770483 \sin (b)$

Step 5

Solve the equation.

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Step 5.1

Rewrite the equation as $57.58770483 \sin (b) = 9$ .

$57.58770483 \sin (b) = 9$

Step 5.2

Divide each term in $57.58770483 \sin (b) = 9$ by $57.58770483$ and simplify.

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Step 5.2.1

Divide each term in $57.58770483 \sin (b) = 9$ by $57.58770483$ .

$\frac{57.58770483 \sin (b)}{57.58770483} = \frac{9}{57.58770483}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $57.58770483$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{57.58770483 \sin (b)}{57.58770483} = \frac{9}{57.58770483}$

Step 5.2.2.1.2

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

$\sin (b) = \frac{9}{57.58770483}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $9$ by $57.58770483$ .

$\sin (b) = 0.15628335$

Step 6

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

$b = \arcsin (0.15628335)$

Step 7

Simplify the right side.

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Step 7.1

Evaluate $\arcsin (0.15628335)$ .

$b = 8.9912345$

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.

$b = 180 - 8.9912345$

Step 9

Subtract $8.9912345$ from $180$ .

$b = 171.00876549$

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.

$b = 8.9912345 + 360 n, 171.00876549 + 360 n$ , for any integer $n$