Precalculus Examples

$(\frac{3 \sqrt{13}}{13}, - \frac{2 \sqrt{13}}{13})$

Step 1

To find the $\tan (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{3 \sqrt{13}}{13}, - \frac{2 \sqrt{13}}{13})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{3 \sqrt{13}}{13}, 0)$ , and $(\frac{3 \sqrt{13}}{13}, - \frac{2 \sqrt{13}}{13})$ .

Opposite : $- \frac{2 \sqrt{13}}{13}$

Adjacent : $\frac{3 \sqrt{13}}{13}$

Step 2

$\tan (θ) = \frac{Opposite}{Adjacent}$ therefore $\tan (θ) = \frac{- \frac{2 \sqrt{13}}{13}}{\frac{3 \sqrt{13}}{13}}$ .

$\frac{- \frac{2 \sqrt{13}}{13}}{\frac{3 \sqrt{13}}{13}}$

Step 3

Simplify $\tan (θ)$ .

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

$\tan (θ) = - \frac{2 \sqrt{13}}{13} \cdot \frac{13}{3 \sqrt{13}}$

Step 3.2

Cancel the common factor of $\sqrt{13}$ .

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Step 3.2.1

Move the leading negative in $- \frac{2 \sqrt{13}}{13}$ into the numerator.

$\tan (θ) = \frac{- 2 \sqrt{13}}{13} \cdot \frac{13}{3 \sqrt{13}}$

Step 3.2.2

Factor $\sqrt{13}$ out of $- 2 \sqrt{13}$ .

$\tan (θ) = \frac{\sqrt{13} \cdot - 2}{13} \cdot \frac{13}{3 \sqrt{13}}$

Step 3.2.3

Factor $\sqrt{13}$ out of $3 \sqrt{13}$ .

$\tan (θ) = \frac{\sqrt{13} \cdot - 2}{13} \cdot \frac{13}{\sqrt{13} \cdot 3}$

Step 3.2.4

Cancel the common factor.

$\tan (θ) = \frac{\sqrt{13} \cdot - 2}{13} \cdot \frac{13}{\sqrt{13} \cdot 3}$

Step 3.2.5

Rewrite the expression.

$\tan (θ) = \frac{- 2}{13} \cdot \frac{13}{3}$

Step 3.3

Cancel the common factor of $13$ .

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Step 3.3.1

Cancel the common factor.

$\tan (θ) = \frac{- 2}{13} \cdot \frac{13}{3}$

Step 3.3.2

Rewrite the expression.

$\tan (θ) = - 2 (\frac{1}{3})$

Step 3.4

Combine $- 2$ and $\frac{1}{3}$ .

$\tan (θ) = \frac{- 2}{3}$

Step 3.5

Move the negative in front of the fraction.

$\tan (θ) = - \frac{2}{3}$

Step 4

Approximate the result.

$\tan (θ) = - \frac{2}{3} \approx - 0.\overline{6}$