Trigonometry Examples

$(\frac{\sqrt{3}}{4}, - \frac{\sqrt{13}}{4})$

Step 1

To find the $\tan (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{\sqrt{3}}{4}, - \frac{\sqrt{13}}{4})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{\sqrt{3}}{4}, 0)$ , and $(\frac{\sqrt{3}}{4}, - \frac{\sqrt{13}}{4})$ .

Opposite : $- \frac{\sqrt{13}}{4}$

Adjacent : $\frac{\sqrt{3}}{4}$

Step 2

$\tan (θ) = \frac{Opposite}{Adjacent}$ therefore $\tan (θ) = \frac{- \frac{\sqrt{13}}{4}}{\frac{\sqrt{3}}{4}}$ .

$\frac{- \frac{\sqrt{13}}{4}}{\frac{\sqrt{3}}{4}}$

Step 3

Simplify $\tan (θ)$ .

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

$\tan (θ) = - \frac{\sqrt{13}}{4} \cdot \frac{4}{\sqrt{3}}$

Step 3.2

Cancel the common factor of $4$ .

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Step 3.2.1

Move the leading negative in $- \frac{\sqrt{13}}{4}$ into the numerator.

$\tan (θ) = \frac{- \sqrt{13}}{4} \cdot \frac{4}{\sqrt{3}}$

Step 3.2.2

Cancel the common factor.

$\tan (θ) = \frac{- \sqrt{13}}{4} \cdot \frac{4}{\sqrt{3}}$

Step 3.2.3

Rewrite the expression.

$\tan (θ) = - \sqrt{13} \frac{1}{\sqrt{3}}$

Step 3.3

Combine $\frac{1}{\sqrt{3}}$ and $\sqrt{13}$ .

$\tan (θ) = - \frac{\sqrt{13}}{\sqrt{3}}$

Step 3.4

Multiply $\frac{\sqrt{13}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\tan (θ) = - (\frac{\sqrt{13}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 3.5

Combine and simplify the denominator.

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Step 3.5.1

Multiply $\frac{\sqrt{13}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\tan (θ) = - \frac{\sqrt{13} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 3.5.2

Raise $\sqrt{3}$ to the power of $1$ .

$\tan (θ) = - \frac{\sqrt{13} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 3.5.3

Raise $\sqrt{3}$ to the power of $1$ .

$\tan (θ) = - \frac{\sqrt{13} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 3.5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\tan (θ) = - \frac{\sqrt{13} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 3.5.5

Add $1$ and $1$ .

$\tan (θ) = - \frac{\sqrt{13} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 3.5.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\tan (θ) = - \frac{\sqrt{13} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 3.5.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\tan (θ) = - \frac{\sqrt{13} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 3.5.6.3

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

$\tan (θ) = - \frac{\sqrt{13} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 3.5.6.4

Cancel the common factor of $2$ .

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Step 3.5.6.4.1

Cancel the common factor.

$\tan (θ) = - \frac{\sqrt{13} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 3.5.6.4.2

Rewrite the expression.

$\tan (θ) = - \frac{\sqrt{13} \sqrt{3}}{3}$

Step 3.5.6.5

Evaluate the exponent.

$\tan (θ) = - \frac{\sqrt{13} \sqrt{3}}{3}$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Combine using the product rule for radicals.

$\tan (θ) = - \frac{\sqrt{13 \cdot 3}}{3}$

Step 3.6.2

Multiply $13$ by $3$ .

$\tan (θ) = - \frac{\sqrt{39}}{3}$

Step 4

Approximate the result.

$\tan (θ) = - \frac{\sqrt{39}}{3} \approx - 2.08166599$