Trigonometry Examples

$(\frac{\sqrt{10}}{10}, - \frac{3 \sqrt{10}}{10})$

Step 1

To find the $\cot (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{\sqrt{10}}{10}, - \frac{3 \sqrt{10}}{10})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{\sqrt{10}}{10}, 0)$ , and $(\frac{\sqrt{10}}{10}, - \frac{3 \sqrt{10}}{10})$ .

Opposite : $- \frac{3 \sqrt{10}}{10}$

Adjacent : $\frac{\sqrt{10}}{10}$

Step 2

$\cot (θ) = \frac{Adjacent}{Opposite}$ therefore $\cot (θ) = \frac{\frac{\sqrt{10}}{10}}{- \frac{3 \sqrt{10}}{10}}$ .

$\frac{\frac{\sqrt{10}}{10}}{- \frac{3 \sqrt{10}}{10}}$

Step 3

Simplify $\cot (θ)$ .

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

$\cot (θ) = \frac{\sqrt{10}}{10} \cdot (- \frac{10}{3 \sqrt{10}})$

Step 3.2

Cancel the common factor of $\sqrt{10}$ .

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Step 3.2.1

Move the leading negative in $- \frac{10}{3 \sqrt{10}}$ into the numerator.

$\cot (θ) = \frac{\sqrt{10}}{10} \cdot \frac{- 10}{3 \sqrt{10}}$

Step 3.2.2

Factor $\sqrt{10}$ out of $3 \sqrt{10}$ .

$\cot (θ) = \frac{\sqrt{10}}{10} \cdot \frac{- 10}{\sqrt{10} \cdot 3}$

Step 3.2.3

Cancel the common factor.

$\cot (θ) = \frac{\sqrt{10}}{10} \cdot \frac{- 10}{\sqrt{10} \cdot 3}$

Step 3.2.4

Rewrite the expression.

$\cot (θ) = \frac{1}{10} \cdot \frac{- 10}{3}$

Step 3.3

Cancel the common factor of $10$ .

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Step 3.3.1

Factor $10$ out of $- 10$ .

$\cot (θ) = \frac{1}{10} \cdot \frac{10 (- 1)}{3}$

Step 3.3.2

Cancel the common factor.

$\cot (θ) = \frac{1}{10} \cdot \frac{10 \cdot - 1}{3}$

Step 3.3.3

Rewrite the expression.

$\cot (θ) = \frac{- 1}{3}$

Step 3.4

Move the negative in front of the fraction.

$\cot (θ) = - \frac{1}{3}$

Step 4

Approximate the result.

$\cot (θ) = - \frac{1}{3} \approx - 0.\overline{3}$