Trigonometry Examples

$\csc^{5} (x) - 4 \csc (x) = 0$

Step 1

Factor $\csc^{5} (x) - 4 \csc (x)$ .

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Step 1.1

Factor $\csc (x)$ out of $\csc^{5} (x) - 4 \csc (x)$ .

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Step 1.1.1

Factor $\csc (x)$ out of $\csc^{5} (x)$ .

$\csc (x) \csc^{4} (x) - 4 \csc (x) = 0$

Step 1.1.2

Factor $\csc (x)$ out of $- 4 \csc (x)$ .

$\csc (x) \csc^{4} (x) + \csc (x) \cdot - 4 = 0$

Step 1.1.3

Factor $\csc (x)$ out of $\csc (x) \csc^{4} (x) + \csc (x) \cdot - 4$ .

$\csc (x) (\csc^{4} (x) - 4) = 0$

Step 1.2

Rewrite $\csc^{4} (x)$ as ${(\csc^{2} (x))}^{2}$ .

$\csc (x) ({(\csc^{2} (x))}^{2} - 4) = 0$

Step 1.3

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

$\csc (x) ({(\csc^{2} (x))}^{2} - 2^{2}) = 0$

Step 1.4

Factor.

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Step 1.4.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \csc^{2} (x)$ and $b = 2$ .

$\csc (x) ((\csc^{2} (x) + 2) (\csc^{2} (x) - 2)) = 0$

Step 1.4.2

Remove unnecessary parentheses.

$\csc (x) (\csc^{2} (x) + 2) (\csc^{2} (x) - 2) = 0$

Step 2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\csc (x) = 0$

$\csc^{2} (x) + 2 = 0$

$\csc^{2} (x) - 2 = 0$

Step 3

Set $\csc (x)$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $\csc (x)$ equal to $0$ .

$\csc (x) = 0$

Step 3.2

The range of cosecant is $y \leq - 1$ and $y \geq 1$ . Since $0$ does not fall in this range, there is no solution.

No solution

Step 4

Set $\csc^{2} (x) - 2$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $\csc^{2} (x) - 2$ equal to $0$ .

$\csc^{2} (x) - 2 = 0$

Step 4.2

Solve $\csc^{2} (x) - 2 = 0$ for $x$ .

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Step 4.2.1

Add $2$ to both sides of the equation.

$\csc^{2} (x) = 2$

Step 4.2.2

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

$\csc (x) = \pm \sqrt{2}$

Step 4.2.3

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

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Step 4.2.3.1

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

$\csc (x) = \sqrt{2}$

Step 4.2.3.2

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

$\csc (x) = - \sqrt{2}$

Step 4.2.3.3

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

$\csc (x) = \sqrt{2}, - \sqrt{2}$

Step 4.2.4

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

$\csc (x) = \sqrt{2}$

$\csc (x) = - \sqrt{2}$

Step 4.2.5

Solve for $x$ in $\csc (x) = \sqrt{2}$ .

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Step 4.2.5.1

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

$x = arccsc (\sqrt{2})$

Step 4.2.5.2

Simplify the right side.

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Step 4.2.5.2.1

The exact value of $arccsc (\sqrt{2})$ is $\frac{π}{4}$ .

$x = \frac{π}{4}$

Step 4.2.5.3

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.

$x = π - \frac{π}{4}$

Step 4.2.5.4

Simplify $π - \frac{π}{4}$ .

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Step 4.2.5.4.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = π \cdot \frac{4}{4} - \frac{π}{4}$

Step 4.2.5.4.2

Combine fractions.

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Step 4.2.5.4.2.1

Combine $π$ and $\frac{4}{4}$ .

$x = \frac{π \cdot 4}{4} - \frac{π}{4}$

Step 4.2.5.4.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 4 - π}{4}$

Step 4.2.5.4.3

Simplify the numerator.

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Step 4.2.5.4.3.1

Move $4$ to the left of $π$ .

$x = \frac{4 \cdot π - π}{4}$

Step 4.2.5.4.3.2

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{4}$

Step 4.2.5.5

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

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Step 4.2.5.5.1

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

$\frac{2 π}{| b |}$

Step 4.2.5.5.2

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

$\frac{2 π}{| 1 |}$

Step 4.2.5.5.3

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

$\frac{2 π}{1}$

Step 4.2.5.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 4.2.5.6

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

$x = \frac{π}{4} + 2 π n, \frac{3 π}{4} + 2 π n$ , for any integer $n$

Step 4.2.6

Solve for $x$ in $\csc (x) = - \sqrt{2}$ .

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Step 4.2.6.1

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

$x = arccsc (- \sqrt{2})$

Step 4.2.6.2

Simplify the right side.

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Step 4.2.6.2.1

The exact value of $arccsc (- \sqrt{2})$ is $- \frac{π}{4}$ .

$x = - \frac{π}{4}$

Step 4.2.6.3

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.

$x = 2 π + \frac{π}{4} + π$

Step 4.2.6.4

Simplify the expression to find the second solution.

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Step 4.2.6.4.1

Subtract $2 π$ from $2 π + \frac{π}{4} + π$ .

$x = 2 π + \frac{π}{4} + π - 2 π$

Step 4.2.6.4.2

The resulting angle of $\frac{5 π}{4}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{4} + π$ .

$x = \frac{5 π}{4}$

Step 4.2.6.5

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

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Step 4.2.6.5.1

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

$\frac{2 π}{| b |}$

Step 4.2.6.5.2

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

$\frac{2 π}{| 1 |}$

Step 4.2.6.5.3

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

$\frac{2 π}{1}$

Step 4.2.6.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 4.2.6.6

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

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Step 4.2.6.6.1

Add $2 π$ to $- \frac{π}{4}$ to find the positive angle.

$- \frac{π}{4} + 2 π$

Step 4.2.6.6.2

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$2 π \cdot \frac{4}{4} - \frac{π}{4}$

Step 4.2.6.6.3

Combine fractions.

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Step 4.2.6.6.3.1

Combine $2 π$ and $\frac{4}{4}$ .

$\frac{2 π \cdot 4}{4} - \frac{π}{4}$

Step 4.2.6.6.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 4 - π}{4}$

Step 4.2.6.6.4

Simplify the numerator.

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Step 4.2.6.6.4.1

Multiply $4$ by $2$ .

$\frac{8 π - π}{4}$

Step 4.2.6.6.4.2

Subtract $π$ from $8 π$ .

$\frac{7 π}{4}$

Step 4.2.6.6.5

List the new angles.

$x = \frac{7 π}{4}$

Step 4.2.6.7

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

$x = \frac{5 π}{4} + 2 π n, \frac{7 π}{4} + 2 π n$ , for any integer $n$

Step 4.2.7

List all of the solutions.

$x = \frac{π}{4} + 2 π n, \frac{3 π}{4} + 2 π n, \frac{5 π}{4} + 2 π n, \frac{7 π}{4} + 2 π n$ , for any integer $n$

Step 4.2.8

Consolidate the answers.

$x = \frac{π}{4} + \frac{π n}{2}$ , for any integer $n$

Step 5

The final solution is all the values that make $\csc (x) (\csc^{2} (x) + 2) (\csc^{2} (x) - 2) = 0$ true.

$x = \frac{π}{4} + \frac{π n}{2}$ , for any integer $n$