Trigonometry Examples

$4 \csc (θ) + 6 = - 2$

Step 1

Move all terms not containing $\csc (θ)$ to the right side of the equation.

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Step 1.1

Subtract $6$ from both sides of the equation.

$4 \csc (θ) = - 2 - 6$

Step 1.2

Subtract $6$ from $- 2$ .

$4 \csc (θ) = - 8$

Step 2

Divide each term in $4 \csc (θ) = - 8$ by $4$ and simplify.

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Step 2.1

Divide each term in $4 \csc (θ) = - 8$ by $4$ .

$\frac{4 \csc (θ)}{4} = \frac{- 8}{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $4$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{4 \csc (θ)}{4} = \frac{- 8}{4}$

Step 2.2.1.2

Divide $\csc (θ)$ by $1$ .

$\csc (θ) = \frac{- 8}{4}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $- 8$ by $4$ .

$\csc (θ) = - 2$

Step 3

Take the inverse cosecant of both sides of the equation to extract $θ$ from inside the cosecant.

$θ = arccsc (- 2)$

Step 4

Simplify the right side.

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Step 4.1

The exact value of $arccsc (- 2)$ is $- 30$ .

$θ = - 30$

Step 5

The cosecant function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $360$ , to find a reference angle. Next, add this reference angle to $180$ to find the solution in the third quadrant.

$θ = 360 + 30 + 180$

Step 6

Simplify the expression to find the second solution.

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Step 6.1

Subtract $360 °$ from $360 + 30 + 180 °$ .

$θ = 360 + 30 + 180 ° - 360 °$

Step 6.2

The resulting angle of $210 °$ is positive, less than $360 °$ , and coterminal with $360 + 30 + 180$ .

$θ = 210 °$

Step 7

Find the period of $\csc (θ)$ .

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

$\frac{360}{1}$

Step 7.4

Divide $360$ by $1$ .

$360$

Step 8

Add $360$ to every negative angle to get positive angles.

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Step 8.1

Add $360$ to $- 30$ to find the positive angle.

$- 30 + 360$

Step 8.2

Subtract $30$ from $360$ .

$330$

Step 8.3

List the new angles.

$θ = 330$

Step 9

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

$θ = 210 + 360 n, 330 + 360 n$ , for any integer $n$