Trigonometry Examples

$(\frac{\sqrt{3}}{3}, \frac{\sqrt{6}}{3})$

Step 1

To find the $\tan (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{\sqrt{3}}{3}, \frac{\sqrt{6}}{3})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{\sqrt{3}}{3}, 0)$ , and $(\frac{\sqrt{3}}{3}, \frac{\sqrt{6}}{3})$ .

Opposite : $\frac{\sqrt{6}}{3}$

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

Step 2

$\tan (θ) = \frac{Opposite}{Adjacent}$ therefore $\tan (θ) = \frac{\frac{\sqrt{6}}{3}}{\frac{\sqrt{3}}{3}}$ .

$\frac{\frac{\sqrt{6}}{3}}{\frac{\sqrt{3}}{3}}$

Step 3

Simplify $\tan (θ)$ .

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

$\tan (θ) = \frac{\sqrt{6}}{3} \cdot \frac{3}{\sqrt{3}}$

Step 3.2

Cancel the common factor of $3$ .

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Step 3.2.1

Cancel the common factor.

$\tan (θ) = \frac{\sqrt{6}}{3} \cdot \frac{3}{\sqrt{3}}$

Step 3.2.2

Rewrite the expression.

$\tan (θ) = \sqrt{6} (\frac{1}{\sqrt{3}})$

Step 3.3

Combine $\sqrt{6}$ and $\frac{1}{\sqrt{3}}$ .

$\tan (θ) = \frac{\sqrt{6}}{\sqrt{3}}$

Step 3.4

Combine $\sqrt{6}$ and $\sqrt{3}$ into a single radical.

$\tan (θ) = \sqrt{\frac{6}{3}}$

Step 3.5

Divide $6$ by $3$ .

$\tan (θ) = \sqrt{2}$

Step 4

Approximate the result.

$\tan (θ) = \sqrt{2} \approx 1.41421356$