Trigonometry Examples

$y = \tan^{2} (6 x)$

Step 1

Rewrite the equation as $\tan^{2} (6 x) = y$ .

$\tan^{2} (6 x) = y$

Step 2

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

$\tan (6 x) = \pm \sqrt{y}$

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.

$\tan (6 x) = \sqrt{y}$

Step 3.2

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

$\tan (6 x) = - \sqrt{y}$

Step 3.3

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

$\tan (6 x) = \sqrt{y}, - \sqrt{y}$

Step 4

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

$\tan (6 x) = \sqrt{y}$

$\tan (6 x) = - \sqrt{y}$

Step 5

Solve for $x$ in $\tan (6 x) = \sqrt{y}$ .

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Step 5.1

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

$6 x = \arctan (\sqrt{y})$

Step 5.2

Divide each term in $6 x = \arctan (\sqrt{y})$ by $6$ and simplify.

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Step 5.2.1

Divide each term in $6 x = \arctan (\sqrt{y})$ by $6$ .

$\frac{6 x}{6} = \frac{\arctan (\sqrt{y})}{6}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $6$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{\arctan (\sqrt{y})}{6}$

Step 5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\arctan (\sqrt{y})}{6}$

Step 6

Solve for $x$ in $\tan (6 x) = - \sqrt{y}$ .

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Step 6.1

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

$6 x = \arctan (- \sqrt{y})$

Step 6.2

Divide each term in $6 x = \arctan (- \sqrt{y})$ by $6$ and simplify.

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Step 6.2.1

Divide each term in $6 x = \arctan (- \sqrt{y})$ by $6$ .

$\frac{6 x}{6} = \frac{\arctan (- \sqrt{y})}{6}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $6$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{\arctan (- \sqrt{y})}{6}$

Step 6.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\arctan (- \sqrt{y})}{6}$

Step 7

List all of the solutions.

$x = \frac{\arctan (\sqrt{y})}{6}$

$x = \frac{\arctan (- \sqrt{y})}{6}$