Trigonometry Examples

$(- \frac{1}{3}, \frac{1}{2})$

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- \frac{1}{3}, \frac{1}{2})$ , draw the triangle between the three points $(0, 0)$ , $(- \frac{1}{3}, 0)$ , and $(- \frac{1}{3}, \frac{1}{2})$ .

Opposite : $\frac{1}{2}$

Adjacent : $- \frac{1}{3}$

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 $- \frac{1}{3}$ .

$\sqrt{{(- 1)}^{2} {(\frac{1}{3})}^{2} + {(\frac{1}{2})}^{2}}$

Step 2.1.2

Apply the product rule to $\frac{1}{3}$ .

$\sqrt{{(- 1)}^{2} \frac{1^{2}}{3^{2}} + {(\frac{1}{2})}^{2}}$

Step 2.2

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

$\sqrt{1 \frac{1^{2}}{3^{2}} + {(\frac{1}{2})}^{2}}$

Step 2.3

Multiply $\frac{1^{2}}{3^{2}}$ by $1$ .

$\sqrt{\frac{1^{2}}{3^{2}} + {(\frac{1}{2})}^{2}}$

Step 2.4

One to any power is one.

$\sqrt{\frac{1}{3^{2}} + {(\frac{1}{2})}^{2}}$

Step 2.5

Raise $3$ to the power of $2$ .

$\sqrt{\frac{1}{9} + {(\frac{1}{2})}^{2}}$

Step 2.6

Apply the product rule to $\frac{1}{2}$ .

$\sqrt{\frac{1}{9} + \frac{1^{2}}{2^{2}}}$

Step 2.7

One to any power is one.

$\sqrt{\frac{1}{9} + \frac{1}{2^{2}}}$

Step 2.8

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

$\sqrt{\frac{1}{9} + \frac{1}{4}}$

Step 2.9

To write $\frac{1}{9}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\sqrt{\frac{1}{9} \cdot \frac{4}{4} + \frac{1}{4}}$

Step 2.10

To write $\frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\sqrt{\frac{1}{9} \cdot \frac{4}{4} + \frac{1}{4} \cdot \frac{9}{9}}$

Step 2.11

Write each expression with a common denominator of $36$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.11.1

Multiply $\frac{1}{9}$ by $\frac{4}{4}$ .

$\sqrt{\frac{4}{9 \cdot 4} + \frac{1}{4} \cdot \frac{9}{9}}$

Step 2.11.2

Multiply $9$ by $4$ .

$\sqrt{\frac{4}{36} + \frac{1}{4} \cdot \frac{9}{9}}$

Step 2.11.3

Multiply $\frac{1}{4}$ by $\frac{9}{9}$ .

$\sqrt{\frac{4}{36} + \frac{9}{4 \cdot 9}}$

Step 2.11.4

Multiply $4$ by $9$ .

$\sqrt{\frac{4}{36} + \frac{9}{36}}$

Step 2.12

Combine the numerators over the common denominator.

$\sqrt{\frac{4 + 9}{36}}$

Step 2.13

Add $4$ and $9$ .

$\sqrt{\frac{13}{36}}$

Step 2.14

Rewrite $\sqrt{\frac{13}{36}}$ as $\frac{\sqrt{13}}{\sqrt{36}}$ .

$\frac{\sqrt{13}}{\sqrt{36}}$

Step 2.15

Simplify the denominator.

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Step 2.15.1

Rewrite $36$ as $6^{2}$ .

$\frac{\sqrt{13}}{\sqrt{6^{2}}}$

Step 2.15.2

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

$\frac{\sqrt{13}}{6}$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{- \frac{1}{3}}{\frac{\sqrt{13}}{6}}$ .

$\frac{- \frac{1}{3}}{\frac{\sqrt{13}}{6}}$

Step 4

Simplify $\cos (θ)$ .

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\cos (θ) = - \frac{1}{3} \cdot \frac{6}{\sqrt{13}}$

Step 4.2

Cancel the common factor of $3$ .

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Step 4.2.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$\cos (θ) = \frac{- 1}{3} \cdot \frac{6}{\sqrt{13}}$

Step 4.2.2

Factor $3$ out of $6$ .

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

Step 4.2.3

Cancel the common factor.

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

Step 4.2.4

Rewrite the expression.

$\cos (θ) = - \frac{2}{\sqrt{13}}$

Step 4.3

Multiply $\frac{2}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\cos (θ) = - (\frac{2}{\sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}})$

Step 4.4

Combine and simplify the denominator.

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Step 4.4.1

Multiply $\frac{2}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\cos (θ) = - \frac{2 \sqrt{13}}{\sqrt{13} \sqrt{13}}$

Step 4.4.2

Raise $\sqrt{13}$ to the power of $1$ .

$\cos (θ) = - \frac{2 \sqrt{13}}{\sqrt{13} \sqrt{13}}$

Step 4.4.3

Raise $\sqrt{13}$ to the power of $1$ .

$\cos (θ) = - \frac{2 \sqrt{13}}{\sqrt{13} \sqrt{13}}$

Step 4.4.4

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

$\cos (θ) = - \frac{2 \sqrt{13}}{{\sqrt{13}}^{1 + 1}}$

Step 4.4.5

Add $1$ and $1$ .

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

Step 4.4.6

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 4.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

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

Step 4.4.6.2

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

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

Step 4.4.6.3

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

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

Step 4.4.6.4

Cancel the common factor of $2$ .

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Step 4.4.6.4.1

Cancel the common factor.

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

Step 4.4.6.4.2

Rewrite the expression.

$\cos (θ) = - \frac{2 \sqrt{13}}{13}$

Step 4.4.6.5

Evaluate the exponent.

$\cos (θ) = - \frac{2 \sqrt{13}}{13}$

Step 5

Approximate the result.

$\cos (θ) = - \frac{2 \sqrt{13}}{13} \approx - 0.55470019$