Precalculus Examples

$\tan (x) = - \frac{2}{\sqrt{5}}$

Step 1

Simplify $- \frac{2}{\sqrt{5}}$ .

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Step 1.1

Multiply $\frac{2}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\tan (x) = - (\frac{2}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}})$

Step 1.2

Combine and simplify the denominator.

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Step 1.2.1

Multiply $\frac{2}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\tan (x) = - \frac{2 \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 1.2.2

Raise $\sqrt{5}$ to the power of $1$ .

$\tan (x) = - \frac{2 \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 1.2.3

Raise $\sqrt{5}$ to the power of $1$ .

$\tan (x) = - \frac{2 \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 1.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\tan (x) = - \frac{2 \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 1.2.5

Add $1$ and $1$ .

$\tan (x) = - \frac{2 \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 1.2.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 1.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\tan (x) = - \frac{2 \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 1.2.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\tan (x) = - \frac{2 \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 1.2.6.3

Combine $\frac{1}{2}$ and $2$ .

$\tan (x) = - \frac{2 \sqrt{5}}{5^{\frac{2}{2}}}$

Step 1.2.6.4

Cancel the common factor of $2$ .

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Step 1.2.6.4.1

Cancel the common factor.

$\tan (x) = - \frac{2 \sqrt{5}}{5^{\frac{2}{2}}}$

Step 1.2.6.4.2

Rewrite the expression.

$\tan (x) = - \frac{2 \sqrt{5}}{5^{1}}$

Step 1.2.6.5

Evaluate the exponent.

$\tan (x) = - \frac{2 \sqrt{5}}{5}$

Step 2

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

$x = \arctan (- \frac{2 \sqrt{5}}{5})$

Step 3

Simplify the right side.

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Step 3.1

Evaluate $\arctan (- \frac{2 \sqrt{5}}{5})$ .

$x = - 0.72972765$

Step 4

The tangent function is negative in the second and fourth quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the third quadrant.

$x = - 0.72972765 - (3.14159265)$

Step 5

Simplify the expression to find the second solution.

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Step 5.1

Add $2 π$ to $- 0.72972765 - (3.14159265)$ .

$x = - 0.72972765 - (3.14159265) + 2 π$

Step 5.2

The resulting angle of $2.41186499$ is positive and coterminal with $- 0.72972765 - (3.14159265)$ .

$x = 2.41186499$

Step 6

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

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Step 6.1

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

$\frac{π}{| b |}$

Step 6.2

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

$\frac{π}{| 1 |}$

Step 6.3

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

$\frac{π}{1}$

Step 6.4

Divide $π$ by $1$ .

$π$

Step 7

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

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Step 7.1

Add $π$ to $- 0.72972765$ to find the positive angle.

$- 0.72972765 + π$

Step 7.2

Replace with decimal approximation.

$3.14159265 - 0.72972765$

Step 7.3

Subtract $0.72972765$ from $3.14159265$ .

$2.41186499$

Step 7.4

List the new angles.

$x = 2.41186499$

Step 8

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

$x = 2.41186499 + π n, 2.41186499 + π n$ , for any integer $n$

Step 9

Consolidate $2.41186499 + π n$ and $2.41186499 + π n$ to $2.41186499 + π n$ .

$x = 2.41186499 + π n$ , for any integer $n$