Trigonometry Examples

$(- \frac{\sqrt{15}}{4}, \frac{1}{4})$

Step 1

To find the $\tan (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- \frac{\sqrt{15}}{4}, \frac{1}{4})$ , draw the triangle between the three points $(0, 0)$ , $(- \frac{\sqrt{15}}{4}, 0)$ , and $(- \frac{\sqrt{15}}{4}, \frac{1}{4})$ .

Opposite : $\frac{1}{4}$

Adjacent : $- \frac{\sqrt{15}}{4}$

Step 2

$\tan (θ) = \frac{Opposite}{Adjacent}$ therefore $\tan (θ) = \frac{\frac{1}{4}}{- \frac{\sqrt{15}}{4}}$ .

$\frac{\frac{1}{4}}{- \frac{\sqrt{15}}{4}}$

Step 3

Simplify $\tan (θ)$ .

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

$\tan (θ) = \frac{1}{4} \cdot (- \frac{4}{\sqrt{15}})$

Step 3.2

Cancel the common factor of $4$ .

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Step 3.2.1

Move the leading negative in $- \frac{4}{\sqrt{15}}$ into the numerator.

$\tan (θ) = \frac{1}{4} \cdot \frac{- 4}{\sqrt{15}}$

Step 3.2.2

Factor $4$ out of $- 4$ .

$\tan (θ) = \frac{1}{4} \cdot \frac{4 (- 1)}{\sqrt{15}}$

Step 3.2.3

Cancel the common factor.

$\tan (θ) = \frac{1}{4} \cdot \frac{4 \cdot - 1}{\sqrt{15}}$

Step 3.2.4

Rewrite the expression.

$\tan (θ) = \frac{- 1}{\sqrt{15}}$

Step 3.3

Move the negative in front of the fraction.

$\tan (θ) = - \frac{1}{\sqrt{15}}$

Step 3.4

Multiply $\frac{1}{\sqrt{15}}$ by $\frac{\sqrt{15}}{\sqrt{15}}$ .

$\tan (θ) = - (\frac{1}{\sqrt{15}} \cdot \frac{\sqrt{15}}{\sqrt{15}})$

Step 3.5

Combine and simplify the denominator.

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Step 3.5.1

Multiply $\frac{1}{\sqrt{15}}$ by $\frac{\sqrt{15}}{\sqrt{15}}$ .

$\tan (θ) = - \frac{\sqrt{15}}{\sqrt{15} \sqrt{15}}$

Step 3.5.2

Raise $\sqrt{15}$ to the power of $1$ .

$\tan (θ) = - \frac{\sqrt{15}}{\sqrt{15} \sqrt{15}}$

Step 3.5.3

Raise $\sqrt{15}$ to the power of $1$ .

$\tan (θ) = - \frac{\sqrt{15}}{\sqrt{15} \sqrt{15}}$

Step 3.5.4

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

$\tan (θ) = - \frac{\sqrt{15}}{{\sqrt{15}}^{1 + 1}}$

Step 3.5.5

Add $1$ and $1$ .

$\tan (θ) = - \frac{\sqrt{15}}{{\sqrt{15}}^{2}}$

Step 3.5.6

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

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Step 3.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$\tan (θ) = - \frac{\sqrt{15}}{{(15^{\frac{1}{2}})}^{2}}$

Step 3.5.6.2

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

$\tan (θ) = - \frac{\sqrt{15}}{15^{\frac{1}{2} \cdot 2}}$

Step 3.5.6.3

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

$\tan (θ) = - \frac{\sqrt{15}}{15^{\frac{2}{2}}}$

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.

$\tan (θ) = - \frac{\sqrt{15}}{15^{\frac{2}{2}}}$

Step 3.5.6.4.2

Rewrite the expression.

$\tan (θ) = - \frac{\sqrt{15}}{15}$

Step 3.5.6.5

Evaluate the exponent.

$\tan (θ) = - \frac{\sqrt{15}}{15}$

Step 4

Approximate the result.

$\tan (θ) = - \frac{\sqrt{15}}{15} \approx - 0.25819888$