Trigonometry Examples

$(- \frac{\sqrt{2}}{5}, \frac{\sqrt{23}}{5})$

Step 1

To find the $\cot (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- \frac{\sqrt{2}}{5}, \frac{\sqrt{23}}{5})$ , draw the triangle between the three points $(0, 0)$ , $(- \frac{\sqrt{2}}{5}, 0)$ , and $(- \frac{\sqrt{2}}{5}, \frac{\sqrt{23}}{5})$ .

Opposite : $\frac{\sqrt{23}}{5}$

Adjacent : $- \frac{\sqrt{2}}{5}$

Step 2

$\cot (θ) = \frac{Adjacent}{Opposite}$ therefore $\cot (θ) = \frac{- \frac{\sqrt{2}}{5}}{\frac{\sqrt{23}}{5}}$ .

$\frac{- \frac{\sqrt{2}}{5}}{\frac{\sqrt{23}}{5}}$

Step 3

Simplify $\cot (θ)$ .

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

$\cot (θ) = - \frac{\sqrt{2}}{5} \cdot \frac{5}{\sqrt{23}}$

Step 3.2

Cancel the common factor of $5$ .

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Step 3.2.1

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

$\cot (θ) = \frac{- \sqrt{2}}{5} \cdot \frac{5}{\sqrt{23}}$

Step 3.2.2

Cancel the common factor.

$\cot (θ) = \frac{- \sqrt{2}}{5} \cdot \frac{5}{\sqrt{23}}$

Step 3.2.3

Rewrite the expression.

$\cot (θ) = - \sqrt{2} \frac{1}{\sqrt{23}}$

Step 3.3

Combine $\frac{1}{\sqrt{23}}$ and $\sqrt{2}$ .

$\cot (θ) = - \frac{\sqrt{2}}{\sqrt{23}}$

Step 3.4

Multiply $\frac{\sqrt{2}}{\sqrt{23}}$ by $\frac{\sqrt{23}}{\sqrt{23}}$ .

$\cot (θ) = - (\frac{\sqrt{2}}{\sqrt{23}} \cdot \frac{\sqrt{23}}{\sqrt{23}})$

Step 3.5

Combine and simplify the denominator.

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Step 3.5.1

Multiply $\frac{\sqrt{2}}{\sqrt{23}}$ by $\frac{\sqrt{23}}{\sqrt{23}}$ .

$\cot (θ) = - \frac{\sqrt{2} \sqrt{23}}{\sqrt{23} \sqrt{23}}$

Step 3.5.2

Raise $\sqrt{23}$ to the power of $1$ .

$\cot (θ) = - \frac{\sqrt{2} \sqrt{23}}{\sqrt{23} \sqrt{23}}$

Step 3.5.3

Raise $\sqrt{23}$ to the power of $1$ .

$\cot (θ) = - \frac{\sqrt{2} \sqrt{23}}{\sqrt{23} \sqrt{23}}$

Step 3.5.4

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

$\cot (θ) = - \frac{\sqrt{2} \sqrt{23}}{{\sqrt{23}}^{1 + 1}}$

Step 3.5.5

Add $1$ and $1$ .

$\cot (θ) = - \frac{\sqrt{2} \sqrt{23}}{{\sqrt{23}}^{2}}$

Step 3.5.6

Rewrite ${\sqrt{23}}^{2}$ as $23$ .

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Step 3.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{23}$ as $23^{\frac{1}{2}}$ .

$\cot (θ) = - \frac{\sqrt{2} \sqrt{23}}{{(23^{\frac{1}{2}})}^{2}}$

Step 3.5.6.2

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

$\cot (θ) = - \frac{\sqrt{2} \sqrt{23}}{23^{\frac{1}{2} \cdot 2}}$

Step 3.5.6.3

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

$\cot (θ) = - \frac{\sqrt{2} \sqrt{23}}{23^{\frac{2}{2}}}$

Step 3.5.6.4

Cancel the common factor of $2$ .

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Step 3.5.6.4.1

Cancel the common factor.

$\cot (θ) = - \frac{\sqrt{2} \sqrt{23}}{23^{\frac{2}{2}}}$

Step 3.5.6.4.2

Rewrite the expression.

$\cot (θ) = - \frac{\sqrt{2} \sqrt{23}}{23}$

Step 3.5.6.5

Evaluate the exponent.

$\cot (θ) = - \frac{\sqrt{2} \sqrt{23}}{23}$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Combine using the product rule for radicals.

$\cot (θ) = - \frac{\sqrt{2 \cdot 23}}{23}$

Step 3.6.2

Multiply $2$ by $23$ .

$\cot (θ) = - \frac{\sqrt{46}}{23}$

Step 4

Approximate the result.

$\cot (θ) = - \frac{\sqrt{46}}{23} \approx - 0.29488391$