Trigonometry Examples

$(\frac{\sqrt{2}}{4}, - \frac{\sqrt{14}}{4})$

Step 1

To find the $\tan (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{\sqrt{2}}{4}, - \frac{\sqrt{14}}{4})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{\sqrt{2}}{4}, 0)$ , and $(\frac{\sqrt{2}}{4}, - \frac{\sqrt{14}}{4})$ .

Opposite : $- \frac{\sqrt{14}}{4}$

Adjacent : $\frac{\sqrt{2}}{4}$

Step 2

$\tan (θ) = \frac{Opposite}{Adjacent}$ therefore $\tan (θ) = \frac{- \frac{\sqrt{14}}{4}}{\frac{\sqrt{2}}{4}}$ .

$\frac{- \frac{\sqrt{14}}{4}}{\frac{\sqrt{2}}{4}}$

Step 3

Simplify $\tan (θ)$ .

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

$\tan (θ) = - \frac{\sqrt{14}}{4} \cdot \frac{4}{\sqrt{2}}$

Step 3.2

Cancel the common factor of $4$ .

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Step 3.2.1

Move the leading negative in $- \frac{\sqrt{14}}{4}$ into the numerator.

$\tan (θ) = \frac{- \sqrt{14}}{4} \cdot \frac{4}{\sqrt{2}}$

Step 3.2.2

Cancel the common factor.

$\tan (θ) = \frac{- \sqrt{14}}{4} \cdot \frac{4}{\sqrt{2}}$

Step 3.2.3

Rewrite the expression.

$\tan (θ) = - \sqrt{14} \frac{1}{\sqrt{2}}$

Step 3.3

Combine $\frac{1}{\sqrt{2}}$ and $\sqrt{14}$ .

$\tan (θ) = - \frac{\sqrt{14}}{\sqrt{2}}$

Step 3.4

Combine $\sqrt{14}$ and $\sqrt{2}$ into a single radical.

$\tan (θ) = - \sqrt{\frac{14}{2}}$

Step 3.5

Divide $14$ by $2$ .

$\tan (θ) = - \sqrt{7}$

Step 4

Approximate the result.

$\tan (θ) = - \sqrt{7} \approx - 2.64575131$