Trigonometry Examples

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

Step 1

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

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

Adjacent : $\frac{3 \sqrt{10}}{10}$

Step 2

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

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

Step 3

Simplify $\tan (θ)$ .

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

Cancel the common factor.

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

Step 3.2.5

Rewrite the expression.

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

Step 3.3

Cancel the common factor of $10$ .

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

$\tan (θ) = - \frac{1}{3}$

Step 4

Approximate the result.

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