Trigonometry Examples

$(- \frac{5}{12}, \frac{12}{13})$

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- \frac{5}{12}, \frac{12}{13})$ , draw the triangle between the three points $(0, 0)$ , $(- \frac{5}{12}, 0)$ , and $(- \frac{5}{12}, \frac{12}{13})$ .

Opposite : $\frac{12}{13}$

Adjacent : $- \frac{5}{12}$

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 $- \frac{5}{12}$ .

$\sqrt{{(- 1)}^{2} {(\frac{5}{12})}^{2} + {(\frac{12}{13})}^{2}}$

Step 2.1.2

Apply the product rule to $\frac{5}{12}$ .

$\sqrt{{(- 1)}^{2} \frac{5^{2}}{12^{2}} + {(\frac{12}{13})}^{2}}$

Step 2.2

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

$\sqrt{1 \frac{5^{2}}{12^{2}} + {(\frac{12}{13})}^{2}}$

Step 2.3

Multiply $\frac{5^{2}}{12^{2}}$ by $1$ .

$\sqrt{\frac{5^{2}}{12^{2}} + {(\frac{12}{13})}^{2}}$

Step 2.4

Raise $5$ to the power of $2$ .

$\sqrt{\frac{25}{12^{2}} + {(\frac{12}{13})}^{2}}$

Step 2.5

Raise $12$ to the power of $2$ .

$\sqrt{\frac{25}{144} + {(\frac{12}{13})}^{2}}$

Step 2.6

Apply the product rule to $\frac{12}{13}$ .

$\sqrt{\frac{25}{144} + \frac{12^{2}}{13^{2}}}$

Step 2.7

Raise $12$ to the power of $2$ .

$\sqrt{\frac{25}{144} + \frac{144}{13^{2}}}$

Step 2.8

Raise $13$ to the power of $2$ .

$\sqrt{\frac{25}{144} + \frac{144}{169}}$

Step 2.9

To write $\frac{25}{144}$ as a fraction with a common denominator, multiply by $\frac{169}{169}$ .

$\sqrt{\frac{25}{144} \cdot \frac{169}{169} + \frac{144}{169}}$

Step 2.10

To write $\frac{144}{169}$ as a fraction with a common denominator, multiply by $\frac{144}{144}$ .

$\sqrt{\frac{25}{144} \cdot \frac{169}{169} + \frac{144}{169} \cdot \frac{144}{144}}$

Step 2.11

Write each expression with a common denominator of $24336$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.11.1

Multiply $\frac{25}{144}$ by $\frac{169}{169}$ .

$\sqrt{\frac{25 \cdot 169}{144 \cdot 169} + \frac{144}{169} \cdot \frac{144}{144}}$

Step 2.11.2

Multiply $144$ by $169$ .

$\sqrt{\frac{25 \cdot 169}{24336} + \frac{144}{169} \cdot \frac{144}{144}}$

Step 2.11.3

Multiply $\frac{144}{169}$ by $\frac{144}{144}$ .

$\sqrt{\frac{25 \cdot 169}{24336} + \frac{144 \cdot 144}{169 \cdot 144}}$

Step 2.11.4

Multiply $169$ by $144$ .

$\sqrt{\frac{25 \cdot 169}{24336} + \frac{144 \cdot 144}{24336}}$

Step 2.12

Combine the numerators over the common denominator.

$\sqrt{\frac{25 \cdot 169 + 144 \cdot 144}{24336}}$

Step 2.13

Simplify the numerator.

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Step 2.13.1

Multiply $25$ by $169$ .

$\sqrt{\frac{4225 + 144 \cdot 144}{24336}}$

Step 2.13.2

Multiply $144$ by $144$ .

$\sqrt{\frac{4225 + 20736}{24336}}$

Step 2.13.3

Add $4225$ and $20736$ .

$\sqrt{\frac{24961}{24336}}$

Step 2.14

Rewrite $\sqrt{\frac{24961}{24336}}$ as $\frac{\sqrt{24961}}{\sqrt{24336}}$ .

$\frac{\sqrt{24961}}{\sqrt{24336}}$

Step 2.15

Simplify the denominator.

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Step 2.15.1

Rewrite $24336$ as $156^{2}$ .

$\frac{\sqrt{24961}}{\sqrt{156^{2}}}$

Step 2.15.2

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

$\frac{\sqrt{24961}}{156}$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{- \frac{5}{12}}{\frac{\sqrt{24961}}{156}}$ .

$\frac{- \frac{5}{12}}{\frac{\sqrt{24961}}{156}}$

Step 4

Simplify $\cos (θ)$ .

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\cos (θ) = - \frac{5}{12} \cdot \frac{156}{\sqrt{24961}}$

Step 4.2

Cancel the common factor of $12$ .

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Step 4.2.1

Move the leading negative in $- \frac{5}{12}$ into the numerator.

$\cos (θ) = \frac{- 5}{12} \cdot \frac{156}{\sqrt{24961}}$

Step 4.2.2

Factor $12$ out of $156$ .

$\cos (θ) = \frac{- 5}{12} \cdot \frac{12 (13)}{\sqrt{24961}}$

Step 4.2.3

Cancel the common factor.

$\cos (θ) = \frac{- 5}{12} \cdot \frac{12 \cdot 13}{\sqrt{24961}}$

Step 4.2.4

Rewrite the expression.

$\cos (θ) = - 5 \frac{13}{\sqrt{24961}}$

Step 4.3

Combine $- 5$ and $\frac{13}{\sqrt{24961}}$ .

$\cos (θ) = \frac{- 5 \cdot 13}{\sqrt{24961}}$

Step 4.4

Simplify the expression.

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Step 4.4.1

Multiply $- 5$ by $13$ .

$\cos (θ) = \frac{- 65}{\sqrt{24961}}$

Step 4.4.2

Move the negative in front of the fraction.

$\cos (θ) = - \frac{65}{\sqrt{24961}}$

Step 4.5

Multiply $\frac{65}{\sqrt{24961}}$ by $\frac{\sqrt{24961}}{\sqrt{24961}}$ .

$\cos (θ) = - (\frac{65}{\sqrt{24961}} \cdot \frac{\sqrt{24961}}{\sqrt{24961}})$

Step 4.6

Combine and simplify the denominator.

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Step 4.6.1

Multiply $\frac{65}{\sqrt{24961}}$ by $\frac{\sqrt{24961}}{\sqrt{24961}}$ .

$\cos (θ) = - \frac{65 \sqrt{24961}}{\sqrt{24961} \sqrt{24961}}$

Step 4.6.2

Raise $\sqrt{24961}$ to the power of $1$ .

$\cos (θ) = - \frac{65 \sqrt{24961}}{\sqrt{24961} \sqrt{24961}}$

Step 4.6.3

Raise $\sqrt{24961}$ to the power of $1$ .

$\cos (θ) = - \frac{65 \sqrt{24961}}{\sqrt{24961} \sqrt{24961}}$

Step 4.6.4

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

$\cos (θ) = - \frac{65 \sqrt{24961}}{{\sqrt{24961}}^{1 + 1}}$

Step 4.6.5

Add $1$ and $1$ .

$\cos (θ) = - \frac{65 \sqrt{24961}}{{\sqrt{24961}}^{2}}$

Step 4.6.6

Rewrite ${\sqrt{24961}}^{2}$ as $24961$ .

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Step 4.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{24961}$ as $24961^{\frac{1}{2}}$ .

$\cos (θ) = - \frac{65 \sqrt{24961}}{{(24961^{\frac{1}{2}})}^{2}}$

Step 4.6.6.2

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

$\cos (θ) = - \frac{65 \sqrt{24961}}{24961^{\frac{1}{2} \cdot 2}}$

Step 4.6.6.3

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

$\cos (θ) = - \frac{65 \sqrt{24961}}{24961^{\frac{2}{2}}}$

Step 4.6.6.4

Cancel the common factor of $2$ .

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Step 4.6.6.4.1

Cancel the common factor.

$\cos (θ) = - \frac{65 \sqrt{24961}}{24961^{\frac{2}{2}}}$

Step 4.6.6.4.2

Rewrite the expression.

$\cos (θ) = - \frac{65 \sqrt{24961}}{24961}$

Step 4.6.6.5

Evaluate the exponent.

$\cos (θ) = - \frac{65 \sqrt{24961}}{24961}$

Step 5

Approximate the result.

$\cos (θ) = - \frac{65 \sqrt{24961}}{24961} \approx - 0.41141712$