Find the Cosine Given the Point (11/61,-60/61)

$(\frac{11}{61}, - \frac{60}{61})$

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{11}{61}, - \frac{60}{61})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{11}{61}, 0)$ , and $(\frac{11}{61}, - \frac{60}{61})$ .

Opposite : $- \frac{60}{61}$

Adjacent : $\frac{11}{61}$

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

Apply the product rule to $\frac{11}{61}$ .

$\sqrt{\frac{11^{2}}{61^{2}} + {(- \frac{60}{61})}^{2}}$

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

$\sqrt{\frac{121}{61^{2}} + {(- \frac{60}{61})}^{2}}$

Raise $61$ to the power of $2$ .

$\sqrt{\frac{121}{3721} + {(- \frac{60}{61})}^{2}}$

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

Apply the product rule to $- \frac{60}{61}$ .

$\sqrt{\frac{121}{3721} + {(- 1)}^{2} {(\frac{60}{61})}^{2}}$

Apply the product rule to $\frac{60}{61}$ .

$\sqrt{\frac{121}{3721} + {(- 1)}^{2} \frac{60^{2}}{61^{2}}}$

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

$\sqrt{\frac{121}{3721} + 1 \frac{60^{2}}{61^{2}}}$

Multiply $\frac{60^{2}}{61^{2}}$ by $1$ .

$\sqrt{\frac{121}{3721} + \frac{60^{2}}{61^{2}}}$

Raise $60$ to the power of $2$ .

$\sqrt{\frac{121}{3721} + \frac{3600}{61^{2}}}$

Raise $61$ to the power of $2$ .

$\sqrt{\frac{121}{3721} + \frac{3600}{3721}}$

Combine the numerators over the common denominator.

$\sqrt{\frac{121 + 3600}{3721}}$

Add $121$ and $3600$ .

$\sqrt{\frac{3721}{3721}}$

Divide $3721$ by $3721$ .

$\sqrt{1}$

Any root of $1$ is $1$ .

$1$

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{\frac{11}{61}}{1}$ .

$\frac{\frac{11}{61}}{1}$

Divide $\frac{11}{61}$ by $1$ .

$\cos (θ) = \frac{11}{61}$

Approximate the result.

$\cos (θ) = \frac{11}{61} \approx 0.18032786$

Find the Cosine Given the Point (11/61,-60/61)

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