Trigonometry Examples

$(- \frac{2}{5}, \frac{\sqrt{21}}{5})$

Step 1

To find the $\tan (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- \frac{2}{5}, \frac{\sqrt{21}}{5})$ , draw the triangle between the three points $(0, 0)$ , $(- \frac{2}{5}, 0)$ , and $(- \frac{2}{5}, \frac{\sqrt{21}}{5})$ .

Opposite : $\frac{\sqrt{21}}{5}$

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

Step 2

$\tan (θ) = \frac{Opposite}{Adjacent}$ therefore $\tan (θ) = \frac{\frac{\sqrt{21}}{5}}{- \frac{2}{5}}$ .

$\frac{\frac{\sqrt{21}}{5}}{- \frac{2}{5}}$

Step 3

Simplify $\tan (θ)$ .

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

$\tan (θ) = \frac{\sqrt{21}}{5} \cdot (- \frac{5}{2})$

Step 3.2

Simplify terms.

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Step 3.2.1

Cancel the common factor of $5$ .

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Step 3.2.1.1

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

$\tan (θ) = \frac{\sqrt{21}}{5} \cdot \frac{- 5}{2}$

Step 3.2.1.2

Factor $5$ out of $- 5$ .

$\tan (θ) = \frac{\sqrt{21}}{5} \cdot \frac{5 (- 1)}{2}$

Step 3.2.1.3

Cancel the common factor.

$\tan (θ) = \frac{\sqrt{21}}{5} \cdot \frac{5 \cdot - 1}{2}$

Step 3.2.1.4

Rewrite the expression.

$\tan (θ) = \sqrt{21} (\frac{- 1}{2})$

Step 3.2.2

Combine $\sqrt{21}$ and $\frac{- 1}{2}$ .

$\tan (θ) = \frac{\sqrt{21} \cdot - 1}{2}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Move $- 1$ to the left of $\sqrt{21}$ .

$\tan (θ) = \frac{- 1 \cdot \sqrt{21}}{2}$

Step 3.3.2

Rewrite $- 1 \sqrt{21}$ as $- \sqrt{21}$ .

$\tan (θ) = \frac{- \sqrt{21}}{2}$

Step 3.4

Move the negative in front of the fraction.

$\tan (θ) = - \frac{\sqrt{21}}{2}$

Step 4

Approximate the result.

$\tan (θ) = - \frac{\sqrt{21}}{2} \approx - 2.29128784$