Trigonometry Examples

$\csc^{2} (θ) - 4 = 0$

Step 1

Add $4$ to both sides of the equation.

$\csc^{2} (θ) = 4$

Step 2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$\csc (θ) = \pm \sqrt{4}$

Step 3

Simplify $\pm \sqrt{4}$ .

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Step 3.1

Rewrite $4$ as $2^{2}$ .

$\csc (θ) = \pm \sqrt{2^{2}}$

Step 3.2

Pull terms out from under the radical, assuming positive real numbers.

$\csc (θ) = \pm 2$

Step 4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.1

First, use the positive value of the $\pm$ to find the first solution.

$\csc (θ) = 2$

Step 4.2

Next, use the negative value of the $\pm$ to find the second solution.

$\csc (θ) = - 2$

Step 4.3

The complete solution is the result of both the positive and negative portions of the solution.

$\csc (θ) = 2, - 2$

Step 5

Set up each of the solutions to solve for $θ$ .

$\csc (θ) = 2$

$\csc (θ) = - 2$

Step 6

Solve for $θ$ in $\csc (θ) = 2$ .

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Step 6.1

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

$θ = arccsc (2)$

Step 6.2

Simplify the right side.

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Step 6.2.1

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

$θ = 30$

Step 6.3

The cosecant 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.

$θ = 180 - 30$

Step 6.4

Subtract $30$ from $180$ .

$θ = 150$

Step 6.5

Find the period of $\csc (θ)$ .

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Step 6.5.1

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

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

Step 6.5.2

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

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

Step 6.5.3

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

$\frac{360}{1}$

Step 6.5.4

Divide $360$ by $1$ .

$360$

Step 6.6

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

$θ = 30 + 360 n, 150 + 360 n$ , for any integer $n$

Step 7

Solve for $θ$ in $\csc (θ) = - 2$ .

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Step 7.1

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

$θ = arccsc (- 2)$

Step 7.2

Simplify the right side.

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Step 7.2.1

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

$θ = - 30$

Step 7.3

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 7.4

Simplify the expression to find the second solution.

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Step 7.4.1

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

$θ = 360 + 30 + 180 ° - 360 °$

Step 7.4.2

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

$θ = 210 °$

Step 7.5

Find the period of $\csc (θ)$ .

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Step 7.5.1

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

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

Step 7.5.2

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

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

Step 7.5.3

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

$\frac{360}{1}$

Step 7.5.4

Divide $360$ by $1$ .

$360$

Step 7.6

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

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Step 7.6.1

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

$- 30 + 360$

Step 7.6.2

Subtract $30$ from $360$ .

$330$

Step 7.6.3

List the new angles.

$θ = 330$

Step 7.7

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 $n$

Step 8

List all of the solutions.

$θ = 30 + 360 n, 150 + 360 n, 210 + 360 n, 330 + 360 n$ , for any integer $n$

Step 9

Consolidate the solutions.

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Step 9.1

Consolidate $30 + 360 n$ and $210 + 360 n$ to $30 + 180 n$ .

$θ = 30 + 180 n, 150 + 360 n, 330 + 360 n$ , for any integer $n$

Step 9.2

Consolidate $150 + 360 n$ and $330 + 360 n$ to $150 + 180 n$ .

$θ = 30 + 180 n, 150 + 180 n$ , for any integer $n$