Trigonometry Examples

$\tan^{2} (x) + 3 = 0$

Step 1

Subtract $3$ from both sides of the equation.

$\tan^{2} (x) = - 3$

Step 2

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

$\tan (x) = \pm \sqrt{- 3}$

Step 3

Simplify $\pm \sqrt{- 3}$ .

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Step 3.1

Rewrite $- 3$ as $- 1 (3)$ .

$\tan (x) = \pm \sqrt{- 1 (3)}$

Step 3.2

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$\tan (x) = \pm \sqrt{- 1} \cdot \sqrt{3}$

Step 3.3

Rewrite $\sqrt{- 1}$ as $i$ .

$\tan (x) = \pm i \sqrt{3}$

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.

$\tan (x) = i \sqrt{3}$

Step 4.2

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

$\tan (x) = - i \sqrt{3}$

Step 4.3

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

$\tan (x) = i \sqrt{3}, - i \sqrt{3}$

Step 5

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

$\tan (x) = i \sqrt{3}$

$\tan (x) = - i \sqrt{3}$

Step 6

Solve for $x$ in $\tan (x) = i \sqrt{3}$ .

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Step 6.1

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

$x = \arctan (i \sqrt{3})$

Step 6.2

The inverse tangent of $\arctan (i \sqrt{3})$ is undefined.

Undefined

Step 7

Solve for $x$ in $\tan (x) = - i \sqrt{3}$ .

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Step 7.1

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

$x = \arctan (- i \sqrt{3})$

Step 7.2

The inverse tangent of $\arctan (- i \sqrt{3})$ is undefined.

Undefined

Step 8

List all of the solutions.

No solution