Trigonometry Examples

$(- \frac{2}{3}, \frac{\sqrt{5}}{3})$

Step 1

To find the $\tan (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- \frac{2}{3}, \frac{\sqrt{5}}{3})$ , draw the triangle between the three points $(0, 0)$ , $(- \frac{2}{3}, 0)$ , and $(- \frac{2}{3}, \frac{\sqrt{5}}{3})$ .

Opposite : $\frac{\sqrt{5}}{3}$

Adjacent : $- \frac{2}{3}$

Step 2

$\tan (θ) = \frac{Opposite}{Adjacent}$ therefore $\tan (θ) = \frac{\frac{\sqrt{5}}{3}}{- \frac{2}{3}}$ .

$\frac{\frac{\sqrt{5}}{3}}{- \frac{2}{3}}$

Step 3

Simplify $\tan (θ)$ .

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

$\tan (θ) = \frac{\sqrt{5}}{3} \cdot (- \frac{3}{2})$

Step 3.2

Simplify terms.

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Step 3.2.1

Cancel the common factor of $3$ .

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Step 3.2.1.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$\tan (θ) = \frac{\sqrt{5}}{3} \cdot \frac{- 3}{2}$

Step 3.2.1.2

Factor $3$ out of $- 3$ .

$\tan (θ) = \frac{\sqrt{5}}{3} \cdot \frac{3 (- 1)}{2}$

Step 3.2.1.3

Cancel the common factor.

$\tan (θ) = \frac{\sqrt{5}}{3} \cdot \frac{3 \cdot - 1}{2}$

Step 3.2.1.4

Rewrite the expression.

$\tan (θ) = \sqrt{5} (\frac{- 1}{2})$

Step 3.2.2

Combine $\sqrt{5}$ and $\frac{- 1}{2}$ .

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

Step 3.3

Simplify the numerator.

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Step 3.3.1

Move $- 1$ to the left of $\sqrt{5}$ .

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

Step 3.3.2

Rewrite $- 1 \sqrt{5}$ as $- \sqrt{5}$ .

$\tan (θ) = \frac{- \sqrt{5}}{2}$

Step 3.4

Move the negative in front of the fraction.

$\tan (θ) = - \frac{\sqrt{5}}{2}$

Step 4

Approximate the result.

$\tan (θ) = - \frac{\sqrt{5}}{2} \approx - 1.11803398$