Trigonometry Examples

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

Step 1

To find the $\cot (θ)$ 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

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

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

Step 3

Simplify $\cot (θ)$ .

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

Cancel the common factor.

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

Step 3.2.5

Rewrite the expression.

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

Step 3.3

Cancel the common factor of $13$ .

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Step 3.3.1

Factor $13$ out of $- 13$ .

$\cot (θ) = \frac{3}{13} \cdot \frac{13 (- 1)}{2}$

Step 3.3.2

Cancel the common factor.

$\cot (θ) = \frac{3}{13} \cdot \frac{13 \cdot - 1}{2}$

Step 3.3.3

Rewrite the expression.

$\cot (θ) = 3 (\frac{- 1}{2})$

Step 3.4

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

$\cot (θ) = \frac{3 \cdot - 1}{2}$

Step 3.5

Simplify the expression.

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Step 3.5.1

Multiply $3$ by $- 1$ .

$\cot (θ) = \frac{- 3}{2}$

Step 3.5.2

Move the negative in front of the fraction.

$\cot (θ) = - \frac{3}{2}$

Step 4

Approximate the result.

$\cot (θ) = - \frac{3}{2} \approx - 1.5$