Trigonometry Examples

$(\frac{1}{\sqrt{10}}, - \frac{3}{\sqrt{10}})$

Step 1

To find the $\tan (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{1}{\sqrt{10}}, - \frac{3}{\sqrt{10}})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{1}{\sqrt{10}}, 0)$ , and $(\frac{1}{\sqrt{10}}, - \frac{3}{\sqrt{10}})$ .

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

Adjacent : $\frac{1}{\sqrt{10}}$

Step 2

$\tan (θ) = \frac{Opposite}{Adjacent}$ therefore $\tan (θ) = \frac{- \frac{3}{\sqrt{10}}}{\frac{1}{\sqrt{10}}}$ .

$\frac{- \frac{3}{\sqrt{10}}}{\frac{1}{\sqrt{10}}}$

Step 3

Simplify $\tan (θ)$ .

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

$\tan (θ) = - 3$