Trigonometry Examples

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

Step 1

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

Opposite : $\frac{- 3 \sqrt{13}}{13}$

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

Step 2

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

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

Step 3

Simplify $\tan (θ)$ .

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

Cancel the common factor.

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

Step 3.2.4

Rewrite the expression.

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

Step 3.3

Cancel the common factor of $13$ .

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

$\tan (θ) = - 3 (\frac{1}{- 2})$

Step 3.4

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

$\tan (θ) = \frac{- 3}{- 2}$

Step 3.5

Dividing two negative values results in a positive value.

$\tan (θ) = \frac{3}{2}$

Step 4

Approximate the result.

$\tan (θ) = \frac{3}{2} \approx 1.5$