Find the Cotangent Given the Point (( square root of 2)/3,-( square root of 7)/3)

$(\frac{\sqrt{2}}{3}, - \frac{\sqrt{7}}{3})$

To find the $\cot (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{\sqrt{2}}{3}, - \frac{\sqrt{7}}{3})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{\sqrt{2}}{3}, 0)$ , and $(\frac{\sqrt{2}}{3}, - \frac{\sqrt{7}}{3})$ .

Opposite : $- \frac{\sqrt{7}}{3}$

Adjacent : $\frac{\sqrt{2}}{3}$

$\cot (θ) = \frac{Adjacent}{Opposite}$ therefore $\cot (θ) = \frac{\frac{\sqrt{2}}{3}}{- \frac{\sqrt{7}}{3}}$ .

$\frac{\frac{\sqrt{2}}{3}}{- \frac{\sqrt{7}}{3}}$

Simplify $\cot (θ)$ .

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Multiply the numerator by the reciprocal of the denominator.

$\cot (θ) = \frac{\sqrt{2}}{3} \cdot (- \frac{3}{\sqrt{7}})$

Simplify terms.

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Cancel the common factor of $3$ .

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Move the leading negative in $- \frac{3}{\sqrt{7}}$ into the numerator.

$\cot (θ) = \frac{\sqrt{2}}{3} \cdot \frac{- 3}{\sqrt{7}}$

Factor $3$ out of $- 3$ .

$\cot (θ) = \frac{\sqrt{2}}{3} \cdot \frac{3 (- 1)}{\sqrt{7}}$

Cancel the common factor.

$\cot (θ) = \frac{\sqrt{2}}{3} \cdot \frac{3 \cdot - 1}{\sqrt{7}}$

Rewrite the expression.

$\cot (θ) = \sqrt{2} (\frac{- 1}{\sqrt{7}})$

Combine $\sqrt{2}$ and $\frac{- 1}{\sqrt{7}}$ .

$\cot (θ) = \frac{\sqrt{2} \cdot - 1}{\sqrt{7}}$

Simplify the numerator.

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Move $- 1$ to the left of $\sqrt{2}$ .

$\cot (θ) = \frac{- 1 \cdot \sqrt{2}}{\sqrt{7}}$

Rewrite $- 1 \sqrt{2}$ as $- \sqrt{2}$ .

$\cot (θ) = \frac{- \sqrt{2}}{\sqrt{7}}$

Move the negative in front of the fraction.

$\cot (θ) = - \frac{\sqrt{2}}{\sqrt{7}}$

Multiply $\frac{\sqrt{2}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\cot (θ) = - (\frac{\sqrt{2}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}})$

Combine and simplify the denominator.

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Multiply $\frac{\sqrt{2}}{\sqrt{7}}$ and $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\cot (θ) = - \frac{\sqrt{2} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Raise $\sqrt{7}$ to the power of $1$ .

$\cot (θ) = - \frac{\sqrt{2} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Raise $\sqrt{7}$ to the power of $1$ .

$\cot (θ) = - \frac{\sqrt{2} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

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

$\cot (θ) = - \frac{\sqrt{2} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Add $1$ and $1$ .

$\cot (θ) = - \frac{\sqrt{2} \sqrt{7}}{{\sqrt{7}}^{2}}$

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\cot (θ) = - \frac{\sqrt{2} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

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

$\cot (θ) = - \frac{\sqrt{2} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

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

$\cot (θ) = - \frac{\sqrt{2} \sqrt{7}}{7^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\cot (θ) = - \frac{\sqrt{2} \sqrt{7}}{7^{\frac{2}{2}}}$

Rewrite the expression.

$\cot (θ) = - \frac{\sqrt{2} \sqrt{7}}{7}$

Evaluate the exponent.

$\cot (θ) = - \frac{\sqrt{2} \sqrt{7}}{7}$

Simplify the numerator.

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Combine using the product rule for radicals.

$\cot (θ) = - \frac{\sqrt{2 \cdot 7}}{7}$

Multiply $2$ by $7$ .

$\cot (θ) = - \frac{\sqrt{14}}{7}$

Approximate the result.

$\cot (θ) = - \frac{\sqrt{14}}{7} \approx - 0.53452248$

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