Trigonometry Examples

$\csc^{2} (x) = {(- \frac{1}{5})}^{2} + 1$

Step 1

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

$\csc (x) = \pm \sqrt{{(- \frac{1}{5})}^{2} + 1}$

Step 2

Simplify $\pm \sqrt{{(- \frac{1}{5})}^{2} + 1}$ .

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Step 2.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.1.1

Apply the product rule to $- \frac{1}{5}$ .

$\csc (x) = \pm \sqrt{{(- 1)}^{2} {(\frac{1}{5})}^{2} + 1}$

Step 2.1.2

Apply the product rule to $\frac{1}{5}$ .

$\csc (x) = \pm \sqrt{{(- 1)}^{2} \frac{1^{2}}{5^{2}} + 1}$

Step 2.2

Raise $- 1$ to the power of $2$ .

$\csc (x) = \pm \sqrt{1 \frac{1^{2}}{5^{2}} + 1}$

Step 2.3

Multiply $\frac{1^{2}}{5^{2}}$ by $1$ .

$\csc (x) = \pm \sqrt{\frac{1^{2}}{5^{2}} + 1}$

Step 2.4

One to any power is one.

$\csc (x) = \pm \sqrt{\frac{1}{5^{2}} + 1}$

Step 2.5

Raise $5$ to the power of $2$ .

$\csc (x) = \pm \sqrt{\frac{1}{25} + 1}$

Step 2.6

Write $1$ as a fraction with a common denominator.

$\csc (x) = \pm \sqrt{\frac{1}{25} + \frac{25}{25}}$

Step 2.7

Combine the numerators over the common denominator.

$\csc (x) = \pm \sqrt{\frac{1 + 25}{25}}$

Step 2.8

Add $1$ and $25$ .

$\csc (x) = \pm \sqrt{\frac{26}{25}}$

Step 2.9

Rewrite $\sqrt{\frac{26}{25}}$ as $\frac{\sqrt{26}}{\sqrt{25}}$ .

$\csc (x) = \pm \frac{\sqrt{26}}{\sqrt{25}}$

Step 2.10

Simplify the denominator.

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Step 2.10.1

Rewrite $25$ as $5^{2}$ .

$\csc (x) = \pm \frac{\sqrt{26}}{\sqrt{5^{2}}}$

Step 2.10.2

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

$\csc (x) = \pm \frac{\sqrt{26}}{5}$

Step 3

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

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Step 3.1

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

$\csc (x) = \frac{\sqrt{26}}{5}$

Step 3.2

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

$\csc (x) = - \frac{\sqrt{26}}{5}$

Step 3.3

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

$\csc (x) = \frac{\sqrt{26}}{5}, - \frac{\sqrt{26}}{5}$

Step 4

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

$\csc (x) = \frac{\sqrt{26}}{5}$

$\csc (x) = - \frac{\sqrt{26}}{5}$

Step 5

Solve for $x$ in $\csc (x) = \frac{\sqrt{26}}{5}$ .

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Step 5.1

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

$x = arccsc (\frac{\sqrt{26}}{5})$

Step 5.2

Simplify the right side.

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Step 5.2.1

Evaluate $arccsc (\frac{\sqrt{26}}{5})$ .

$x = 1.37340076$

Step 5.3

The cosecant function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$x = (3.14159265) - 1.37340076$

Step 5.4

Solve for $x$ .

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Step 5.4.1

Remove parentheses.

$x = 3.14159265 - 1.37340076$

Step 5.4.2

Remove parentheses.

$x = (3.14159265) - 1.37340076$

Step 5.4.3

Subtract $1.37340076$ from $3.14159265$ .

$x = 1.76819188$

Step 5.5

Find the period of $\csc (x)$ .

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Step 5.5.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 5.5.2

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

$\frac{2 π}{| 1 |}$

Step 5.5.3

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

$\frac{2 π}{1}$

Step 5.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 5.6

The period of the $\csc (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = 1.37340076 + 2 π n, 1.76819188 + 2 π n$ , for any integer $n$

Step 6

Solve for $x$ in $\csc (x) = - \frac{\sqrt{26}}{5}$ .

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Step 6.1

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

$x = arccsc (- \frac{\sqrt{26}}{5})$

Step 6.2

Simplify the right side.

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Step 6.2.1

Evaluate $arccsc (- \frac{\sqrt{26}}{5})$ .

$x = - 1.37340076$

Step 6.3

The cosecant function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$x = 2 (3.14159265) + 1.37340076 + 3.14159265$

Step 6.4

Simplify the expression to find the second solution.

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Step 6.4.1

Subtract $2 π$ from $2 (3.14159265) + 1.37340076 + 3.14159265$ .

$x = 2 (3.14159265) + 1.37340076 + 3.14159265 - 2 π$

Step 6.4.2

The resulting angle of $4.51499342$ is positive, less than $2 π$ , and coterminal with $2 (3.14159265) + 1.37340076 + 3.14159265$ .

$x = 4.51499342$

Step 6.5

Find the period of $\csc (x)$ .

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Step 6.5.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 6.5.2

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

$\frac{2 π}{| 1 |}$

Step 6.5.3

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

$\frac{2 π}{1}$

Step 6.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 6.6

Add $2 π$ to every negative angle to get positive angles.

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Step 6.6.1

Add $2 π$ to $- 1.37340076$ to find the positive angle.

$- 1.37340076 + 2 π$

Step 6.6.2

Subtract $1.37340076$ from $2 π$ .

$4.90978454$

Step 6.6.3

List the new angles.

$x = 4.90978454$

Step 6.7

The period of the $\csc (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = 4.51499342 + 2 π n, 4.90978454 + 2 π n$ , for any integer $n$

Step 7

List all of the solutions.

$x = 1.37340076 + 2 π n, 1.76819188 + 2 π n, 4.51499342 + 2 π n, 4.90978454 + 2 π n$ , for any integer $n$

Step 8

Consolidate the solutions.

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Step 8.1

Consolidate $1.37340076 + 2 π n$ and $4.51499342 + 2 π n$ to $1.37340076 + π n$ .

$x = 1.37340076 + π n, 1.76819188 + 2 π n, 4.90978454 + 2 π n$ , for any integer $n$

Step 8.2

Consolidate $1.76819188 + 2 π n$ and $4.90978454 + 2 π n$ to $1.76819188 + π n$ .

$x = 1.37340076 + π n, 1.76819188 + π n$ , for any integer $n$