Precalculus Examples

$(\frac{2 \sqrt{6}}{5}, \frac{1}{5})$

Step 1

To find the $\tan (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{2 \sqrt{6}}{5}, \frac{1}{5})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{2 \sqrt{6}}{5}, 0)$ , and $(\frac{2 \sqrt{6}}{5}, \frac{1}{5})$ .

Opposite : $\frac{1}{5}$

Adjacent : $\frac{2 \sqrt{6}}{5}$

Step 2

$\tan (θ) = \frac{Opposite}{Adjacent}$ therefore $\tan (θ) = \frac{\frac{1}{5}}{\frac{2 \sqrt{6}}{5}}$ .

$\frac{\frac{1}{5}}{\frac{2 \sqrt{6}}{5}}$

Step 3

Simplify $\tan (θ)$ .

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

$\tan (θ) = \frac{1}{5} \cdot \frac{5}{2 \sqrt{6}}$

Step 3.2

Cancel the common factor of $5$ .

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Step 3.2.1

Cancel the common factor.

$\tan (θ) = \frac{1}{5} \cdot \frac{5}{2 \sqrt{6}}$

Step 3.2.2

Rewrite the expression.

$\tan (θ) = \frac{1}{2 \sqrt{6}}$

Step 3.3

Multiply $\frac{1}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\tan (θ) = \frac{1}{2 \sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

Step 3.4

Combine and simplify the denominator.

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Step 3.4.1

Multiply $\frac{1}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\tan (θ) = \frac{\sqrt{6}}{2 \sqrt{6} \sqrt{6}}$

Step 3.4.2

Move $\sqrt{6}$ .

$\tan (θ) = \frac{\sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

Step 3.4.3

Raise $\sqrt{6}$ to the power of $1$ .

$\tan (θ) = \frac{\sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

Step 3.4.4

Raise $\sqrt{6}$ to the power of $1$ .

$\tan (θ) = \frac{\sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

Step 3.4.5

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

$\tan (θ) = \frac{\sqrt{6}}{2 {\sqrt{6}}^{1 + 1}}$

Step 3.4.6

Add $1$ and $1$ .

$\tan (θ) = \frac{\sqrt{6}}{2 {\sqrt{6}}^{2}}$

Step 3.4.7

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 3.4.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\tan (θ) = \frac{\sqrt{6}}{2 {(6^{\frac{1}{2}})}^{2}}$

Step 3.4.7.2

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

$\tan (θ) = \frac{\sqrt{6}}{2 \cdot 6^{\frac{1}{2} \cdot 2}}$

Step 3.4.7.3

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

$\tan (θ) = \frac{\sqrt{6}}{2 \cdot 6^{\frac{2}{2}}}$

Step 3.4.7.4

Cancel the common factor of $2$ .

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Step 3.4.7.4.1

Cancel the common factor.

$\tan (θ) = \frac{\sqrt{6}}{2 \cdot 6^{\frac{2}{2}}}$

Step 3.4.7.4.2

Rewrite the expression.

$\tan (θ) = \frac{\sqrt{6}}{2 \cdot 6}$

Step 3.4.7.5

Evaluate the exponent.

$\tan (θ) = \frac{\sqrt{6}}{2 \cdot 6}$

Step 3.5

Multiply $2$ by $6$ .

$\tan (θ) = \frac{\sqrt{6}}{12}$

Step 4

Approximate the result.

$\tan (θ) = \frac{\sqrt{6}}{12} \approx 0.20412414$