Precalculus Examples

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

Step 1

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

Opposite : $\frac{\sqrt{15}}{4}$

Adjacent : $- \frac{1}{4}$

Step 2

Find the hypotenuse using Pythagorean theorem $c = \sqrt{a^{2} + b^{2}}$ .

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Step 2.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.1.1

Apply the product rule to $- \frac{1}{4}$ .

$\sqrt{{(- 1)}^{2} {(\frac{1}{4})}^{2} + {(\frac{\sqrt{15}}{4})}^{2}}$

Step 2.1.2

Apply the product rule to $\frac{1}{4}$ .

$\sqrt{{(- 1)}^{2} \frac{1^{2}}{4^{2}} + {(\frac{\sqrt{15}}{4})}^{2}}$

Step 2.2

Simplify the expression.

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Step 2.2.1

Raise $- 1$ to the power of $2$ .

$\sqrt{1 \frac{1^{2}}{4^{2}} + {(\frac{\sqrt{15}}{4})}^{2}}$

Step 2.2.2

Multiply $\frac{1^{2}}{4^{2}}$ by $1$ .

$\sqrt{\frac{1^{2}}{4^{2}} + {(\frac{\sqrt{15}}{4})}^{2}}$

Step 2.2.3

One to any power is one.

$\sqrt{\frac{1}{4^{2}} + {(\frac{\sqrt{15}}{4})}^{2}}$

Step 2.2.4

Raise $4$ to the power of $2$ .

$\sqrt{\frac{1}{16} + {(\frac{\sqrt{15}}{4})}^{2}}$

Step 2.2.5

Apply the product rule to $\frac{\sqrt{15}}{4}$ .

$\sqrt{\frac{1}{16} + \frac{{\sqrt{15}}^{2}}{4^{2}}}$

Step 2.3

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

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Step 2.3.1

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

$\sqrt{\frac{1}{16} + \frac{{(15^{\frac{1}{2}})}^{2}}{4^{2}}}$

Step 2.3.2

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

$\sqrt{\frac{1}{16} + \frac{15^{\frac{1}{2} \cdot 2}}{4^{2}}}$

Step 2.3.3

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

$\sqrt{\frac{1}{16} + \frac{15^{\frac{2}{2}}}{4^{2}}}$

Step 2.3.4

Cancel the common factor of $2$ .

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Step 2.3.4.1

Cancel the common factor.

$\sqrt{\frac{1}{16} + \frac{15^{\frac{2}{2}}}{4^{2}}}$

Step 2.3.4.2

Rewrite the expression.

$\sqrt{\frac{1}{16} + \frac{15^{1}}{4^{2}}}$

Step 2.3.5

Evaluate the exponent.

$\sqrt{\frac{1}{16} + \frac{15}{4^{2}}}$

Step 2.4

Simplify the expression.

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Step 2.4.1

Raise $4$ to the power of $2$ .

$\sqrt{\frac{1}{16} + \frac{15}{16}}$

Step 2.4.2

Combine the numerators over the common denominator.

$\sqrt{\frac{1 + 15}{16}}$

Step 2.4.3

Add $1$ and $15$ .

$\sqrt{\frac{16}{16}}$

Step 2.4.4

Divide $16$ by $16$ .

$\sqrt{1}$

Step 2.4.5

Any root of $1$ is $1$ .

$1$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{- \frac{1}{4}}{1}$ .

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

Step 4

Divide $- \frac{1}{4}$ by $1$ .

$\cos (θ) = - \frac{1}{4}$

Step 5

Approximate the result.

$\cos (θ) = - \frac{1}{4} \approx - 0.25$