Solve for θ csc(theta)^2-2=0

$\csc^{2} (θ) - 2 = 0$

Add $2$ to both sides of the equation.

$\csc^{2} (θ) = 2$

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

$\csc (θ) = \pm \sqrt{2}$

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

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

$\csc (θ) = \sqrt{2}$

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

$\csc (θ) = - \sqrt{2}$

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

$\csc (θ) = \sqrt{2}, - \sqrt{2}$

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

$\csc (θ) = \sqrt{2}$

$\csc (θ) = - \sqrt{2}$

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

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

$θ = arccsc (\sqrt{2})$

Simplify the right side.

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

$θ = \frac{π}{4}$

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.

$θ = π - \frac{π}{4}$

Simplify $π - \frac{π}{4}$ .

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

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

Combine fractions.

Combine $π$ and $\frac{4}{4}$ .

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

Combine the numerators over the common denominator.

$θ = \frac{π \cdot 4 - π}{4}$

Simplify the numerator.

Move $4$ to the left of $π$ .

$θ = \frac{4 \cdot π - π}{4}$

Subtract $π$ from $4 π$ .

$θ = \frac{3 π}{4}$

Find the period of $\csc (θ)$ .

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| 1 |}$

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

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

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

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

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

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

$θ = arccsc (- \sqrt{2})$

Simplify the right side.

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

$θ = - \frac{π}{4}$

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.

$θ = 2 π + \frac{π}{4} + π$

Simplify the expression to find the second solution.

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

$θ = 2 π + \frac{π}{4} + π - 2 π$

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

$θ = \frac{5 π}{4}$

Find the period of $\csc (θ)$ .

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| 1 |}$

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

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

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

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

$- \frac{π}{4} + 2 π$

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

$2 π \cdot \frac{4}{4} - \frac{π}{4}$

Combine fractions.

Combine $2 π$ and $\frac{4}{4}$ .

$\frac{2 π \cdot 4}{4} - \frac{π}{4}$

Combine the numerators over the common denominator.

$\frac{2 π \cdot 4 - π}{4}$

Simplify the numerator.

Multiply $4$ by $2$ .

$\frac{8 π - π}{4}$

Subtract $π$ from $8 π$ .

$\frac{7 π}{4}$

List the new angles.

$θ = \frac{7 π}{4}$

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

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

List all of the solutions.

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

Consolidate the answers.

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

Solve for θ csc(theta)^2-2=0

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