Find the Cotangent Given the Point (1/5,-(2 square root of 6)/5)

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

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

Opposite : $- \frac{2 \sqrt{6}}{5}$

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

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

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

Simplify $\cot (θ)$ .

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Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $5$ .

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Move the leading negative in $- \frac{5}{2 \sqrt{6}}$ into the numerator.

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

Factor $5$ out of $- 5$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\cot (θ) = \frac{- 1}{2 \sqrt{6}}$

Move the negative in front of the fraction.

$\cot (θ) = - \frac{1}{2 \sqrt{6}}$

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

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

Combine and simplify the denominator.

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

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

Move $\sqrt{6}$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Multiply $2$ by $6$ .

$\cot (θ) = - \frac{\sqrt{6}}{12}$

Approximate the result.

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

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