Trigonometry Examples

$(- \frac{1}{3}, \frac{2 \sqrt{2}}{3})$

Step 1

To find the $\cot (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- \frac{1}{3}, \frac{2 \sqrt{2}}{3})$ , draw the triangle between the three points $(0, 0)$ , $(- \frac{1}{3}, 0)$ , and $(- \frac{1}{3}, \frac{2 \sqrt{2}}{3})$ .

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

Adjacent : $- \frac{1}{3}$

Step 2

$\cot (θ) = \frac{Adjacent}{Opposite}$ therefore $\cot (θ) = \frac{- \frac{1}{3}}{\frac{2 \sqrt{2}}{3}}$ .

$\frac{- \frac{1}{3}}{\frac{2 \sqrt{2}}{3}}$

Step 3

Simplify $\cot (θ)$ .

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.2

Cancel the common factor of $3$ .

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Step 3.2.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

$\cot (θ) = - \frac{1}{2 \sqrt{2}}$

Step 3.3

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

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

Step 3.4

Combine and simplify the denominator.

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Step 3.4.1

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

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

Step 3.4.2

Move $\sqrt{2}$ .

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

Step 3.4.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 3.4.4

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 3.4.5

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

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

Step 3.4.6

Add $1$ and $1$ .

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

Step 3.4.7

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

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Step 3.4.7.1

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

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

Step 3.4.7.2

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

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

Step 3.4.7.3

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

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

Step 3.4.7.4

Cancel the common factor of $2$ .

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Step 3.4.7.4.1

Cancel the common factor.

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

Step 3.4.7.4.2

Rewrite the expression.

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

Step 3.4.7.5

Evaluate the exponent.

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

Step 3.5

Multiply $2$ by $2$ .

$\cot (θ) = - \frac{\sqrt{2}}{4}$

Step 4

Approximate the result.

$\cot (θ) = - \frac{\sqrt{2}}{4} \approx - 0.35355339$