Trigonometry Examples

$y = \cos^{2} (x) - 2$

Step 1

Rewrite the equation as $\cos^{2} (x) - 2 = y$ .

$\cos^{2} (x) - 2 = y$

Step 2

Add $2$ to both sides of the equation.

$\cos^{2} (x) = y + 2$

Step 3

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

$\cos (x) = \pm \sqrt{y + 2}$

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.

$\cos (x) = \sqrt{y + 2}$

Step 4.2

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

$\cos (x) = - \sqrt{y + 2}$

Step 4.3

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

$\cos (x) = \sqrt{y + 2}, - \sqrt{y + 2}$

Step 5

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

$\cos (x) = \sqrt{y + 2}$

$\cos (x) = - \sqrt{y + 2}$

Step 6

Solve for $x$ in $\cos (x) = \sqrt{y + 2}$ .

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Step 6.1

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

$x = \arccos (\sqrt{y + 2})$

Step 7

Solve for $x$ in $\cos (x) = - \sqrt{y + 2}$ .

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Step 7.1

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

$x = \arccos (- \sqrt{y + 2})$

Step 8

List all of the solutions.

$x = \arccos (\sqrt{y + 2})$

$x = \arccos (- \sqrt{y + 2})$