Precalculus Examples

$(5 \frac{1}{2}, - 2 \sqrt{15})$

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(5 \frac{1}{2}, - 2 \sqrt{15})$ , draw the triangle between the three points $(0, 0)$ , $(5 \frac{1}{2}, 0)$ , and $(5 \frac{1}{2}, - 2 \sqrt{15})$ .

Opposite : $- 2 \sqrt{15}$

Adjacent : $5 \frac{1}{2}$

Step 2

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

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Step 2.1

Convert $5 \frac{1}{2}$ to an improper fraction.

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Step 2.1.1

A mixed number is an addition of its whole and fractional parts.

$\sqrt{{(5 + \frac{1}{2})}^{2} + {(- 2 \sqrt{15})}^{2}}$

Step 2.1.2

Add $5$ and $\frac{1}{2}$ .

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Step 2.1.2.1

To write $5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\sqrt{{(5 \cdot \frac{2}{2} + \frac{1}{2})}^{2} + {(- 2 \sqrt{15})}^{2}}$

Step 2.1.2.2

Combine $5$ and $\frac{2}{2}$ .

$\sqrt{{(\frac{5 \cdot 2}{2} + \frac{1}{2})}^{2} + {(- 2 \sqrt{15})}^{2}}$

Step 2.1.2.3

Combine the numerators over the common denominator.

$\sqrt{{(\frac{5 \cdot 2 + 1}{2})}^{2} + {(- 2 \sqrt{15})}^{2}}$

Step 2.1.2.4

Simplify the numerator.

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Step 2.1.2.4.1

Multiply $5$ by $2$ .

$\sqrt{{(\frac{10 + 1}{2})}^{2} + {(- 2 \sqrt{15})}^{2}}$

Step 2.1.2.4.2

Add $10$ and $1$ .

$\sqrt{{(\frac{11}{2})}^{2} + {(- 2 \sqrt{15})}^{2}}$

Step 2.2

Simplify the expression.

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Step 2.2.1

Apply the product rule to $\frac{11}{2}$ .

$\sqrt{\frac{11^{2}}{2^{2}} + {(- 2 \sqrt{15})}^{2}}$

Step 2.2.2

Raise $11$ to the power of $2$ .

$\sqrt{\frac{121}{2^{2}} + {(- 2 \sqrt{15})}^{2}}$

Step 2.2.3

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

$\sqrt{\frac{121}{4} + {(- 2 \sqrt{15})}^{2}}$

Step 2.2.4

Apply the product rule to $- 2 \sqrt{15}$ .

$\sqrt{\frac{121}{4} + {(- 2)}^{2} {\sqrt{15}}^{2}}$

Step 2.2.5

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

$\sqrt{\frac{121}{4} + 4 {\sqrt{15}}^{2}}$

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 $15^{\frac{1}{2}}$ .

$\sqrt{\frac{121}{4} + 4 {(15^{\frac{1}{2}})}^{2}}$

Step 2.3.2

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

$\sqrt{\frac{121}{4} + 4 \cdot 15^{\frac{1}{2} \cdot 2}}$

Step 2.3.3

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

$\sqrt{\frac{121}{4} + 4 \cdot 15^{\frac{2}{2}}}$

Step 2.3.4

Cancel the common factor of $2$ .

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Step 2.3.4.1

Cancel the common factor.

$\sqrt{\frac{121}{4} + 4 \cdot 15^{\frac{2}{2}}}$

Step 2.3.4.2

Rewrite the expression.

$\sqrt{\frac{121}{4} + 4 \cdot 15^{1}}$

Step 2.3.5

Evaluate the exponent.

$\sqrt{\frac{121}{4} + 4 \cdot 15}$

Step 2.4

Multiply $4$ by $15$ .

$\sqrt{\frac{121}{4} + 60}$

Step 2.5

To write $60$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\sqrt{\frac{121}{4} + 60 \cdot \frac{4}{4}}$

Step 2.6

Combine $60$ and $\frac{4}{4}$ .

$\sqrt{\frac{121}{4} + \frac{60 \cdot 4}{4}}$

Step 2.7

Combine the numerators over the common denominator.

$\sqrt{\frac{121 + 60 \cdot 4}{4}}$

Step 2.8

Simplify the numerator.

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Step 2.8.1

Multiply $60$ by $4$ .

$\sqrt{\frac{121 + 240}{4}}$

Step 2.8.2

Add $121$ and $240$ .

$\sqrt{\frac{361}{4}}$

Step 2.9

Rewrite $\sqrt{\frac{361}{4}}$ as $\frac{\sqrt{361}}{\sqrt{4}}$ .

$\frac{\sqrt{361}}{\sqrt{4}}$

Step 2.10

Simplify the numerator.

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Step 2.10.1

Rewrite $361$ as $19^{2}$ .

$\frac{\sqrt{19^{2}}}{\sqrt{4}}$

Step 2.10.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{19}{\sqrt{4}}$

Step 2.11

Simplify the denominator.

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Step 2.11.1

Rewrite $4$ as $2^{2}$ .

$\frac{19}{\sqrt{2^{2}}}$

Step 2.11.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{19}{2}$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{5 \frac{1}{2}}{\frac{19}{2}}$ .

$\frac{5 \frac{1}{2}}{\frac{19}{2}}$

Step 4

Simplify $\cos (θ)$ .

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Step 4.1

Convert $5 \frac{1}{2}$ to an improper fraction.

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Step 4.1.1

A mixed number is an addition of its whole and fractional parts.

$\cos (θ) = \frac{5 + \frac{1}{2}}{\frac{19}{2}}$

Step 4.1.2

Add $5$ and $\frac{1}{2}$ .

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Step 4.1.2.1

To write $5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\cos (θ) = \frac{5 \cdot \frac{2}{2} + \frac{1}{2}}{\frac{19}{2}}$

Step 4.1.2.2

Combine $5$ and $\frac{2}{2}$ .

$\cos (θ) = \frac{\frac{5 \cdot 2}{2} + \frac{1}{2}}{\frac{19}{2}}$

Step 4.1.2.3

Combine the numerators over the common denominator.

$\cos (θ) = \frac{\frac{5 \cdot 2 + 1}{2}}{\frac{19}{2}}$

Step 4.1.2.4

Simplify the numerator.

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Step 4.1.2.4.1

Multiply $5$ by $2$ .

$\cos (θ) = \frac{\frac{10 + 1}{2}}{\frac{19}{2}}$

Step 4.1.2.4.2

Add $10$ and $1$ .

$\cos (θ) = \frac{\frac{11}{2}}{\frac{19}{2}}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$\cos (θ) = \frac{11}{2} \cdot \frac{2}{19}$

Step 4.3

Cancel the common factor of $2$ .

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Step 4.3.1

Cancel the common factor.

$\cos (θ) = \frac{11}{2} \cdot \frac{2}{19}$

Step 4.3.2

Rewrite the expression.

$\cos (θ) = 11 (\frac{1}{19})$

Step 4.4

Combine $11$ and $\frac{1}{19}$ .

$\cos (θ) = \frac{11}{19}$

Step 5

Approximate the result.

$\cos (θ) = \frac{11}{19} \approx 0.57894736$