Precalculus Examples

$(\frac{21}{29}, - \frac{20}{29})$

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{21}{29}, - \frac{20}{29})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{21}{29}, 0)$ , and $(\frac{21}{29}, - \frac{20}{29})$ .

Opposite : $- \frac{20}{29}$

Adjacent : $\frac{21}{29}$

Step 2

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

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Step 2.1

Apply the product rule to $\frac{21}{29}$ .

$\sqrt{\frac{21^{2}}{29^{2}} + {(- \frac{20}{29})}^{2}}$

Step 2.2

Raise $21$ to the power of $2$ .

$\sqrt{\frac{441}{29^{2}} + {(- \frac{20}{29})}^{2}}$

Step 2.3

Raise $29$ to the power of $2$ .

$\sqrt{\frac{441}{841} + {(- \frac{20}{29})}^{2}}$

Step 2.4

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

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Step 2.4.1

Apply the product rule to $- \frac{20}{29}$ .

$\sqrt{\frac{441}{841} + {(- 1)}^{2} {(\frac{20}{29})}^{2}}$

Step 2.4.2

Apply the product rule to $\frac{20}{29}$ .

$\sqrt{\frac{441}{841} + {(- 1)}^{2} \frac{20^{2}}{29^{2}}}$

Step 2.5

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

$\sqrt{\frac{441}{841} + 1 \frac{20^{2}}{29^{2}}}$

Step 2.6

Multiply $\frac{20^{2}}{29^{2}}$ by $1$ .

$\sqrt{\frac{441}{841} + \frac{20^{2}}{29^{2}}}$

Step 2.7

Raise $20$ to the power of $2$ .

$\sqrt{\frac{441}{841} + \frac{400}{29^{2}}}$

Step 2.8

Raise $29$ to the power of $2$ .

$\sqrt{\frac{441}{841} + \frac{400}{841}}$

Step 2.9

Combine the numerators over the common denominator.

$\sqrt{\frac{441 + 400}{841}}$

Step 2.10

Add $441$ and $400$ .

$\sqrt{\frac{841}{841}}$

Step 2.11

Divide $841$ by $841$ .

$\sqrt{1}$

Step 2.12

Any root of $1$ is $1$ .

$1$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{\frac{21}{29}}{1}$ .

$\frac{\frac{21}{29}}{1}$

Step 4

Divide $\frac{21}{29}$ by $1$ .

$\cos (θ) = \frac{21}{29}$

Step 5

Approximate the result.

$\cos (θ) = \frac{21}{29} \approx 0.72413793$