Trigonometry Examples

$(- \frac{1}{2}, \frac{\sqrt{3}}{2})$

Step 1

To find the $\tan (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- \frac{1}{2}, \frac{\sqrt{3}}{2})$ , draw the triangle between the three points $(0, 0)$ , $(- \frac{1}{2}, 0)$ , and $(- \frac{1}{2}, \frac{\sqrt{3}}{2})$ .

Opposite : $\frac{\sqrt{3}}{2}$

Adjacent : $- \frac{1}{2}$

Step 2

$\tan (θ) = \frac{Opposite}{Adjacent}$ therefore $\tan (θ) = \frac{\frac{\sqrt{3}}{2}}{- \frac{1}{2}}$ .

$\frac{\frac{\sqrt{3}}{2}}{- \frac{1}{2}}$

Step 3

Simplify $\tan (θ)$ .

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.2

Cancel the common factor of $2$ .

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Step 3.2.1

Factor $2$ out of $- 1 \cdot 2$ .

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

$\tan (θ) = \sqrt{3} \cdot - 1$

Step 3.3

Simplify the expression.

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Step 3.3.1

Move $- 1$ to the left of $\sqrt{3}$ .

$\tan (θ) = - 1 \cdot \sqrt{3}$

Step 3.3.2

Rewrite $- 1 \sqrt{3}$ as $- \sqrt{3}$ .

$\tan (θ) = - \sqrt{3}$

Step 4

Approximate the result.

$\tan (θ) = - \sqrt{3} \approx - 1.7320508$