Find the Cosine Given the Point (8,3)

$(8, 3)$

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(8, 3)$ , draw the triangle between the three points $(0, 0)$ , $(8, 0)$ , and $(8, 3)$ .

Opposite : $3$

Adjacent : $8$

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

Raise $8$ to the power of $2$ .

$\sqrt{64 + {(3)}^{2}}$

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

$\sqrt{64 + 9}$

Add $64$ and $9$ .

$\sqrt{73}$

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{8}{\sqrt{73}}$ .

$\frac{8}{\sqrt{73}}$

Simplify $\cos (θ)$ .

Multiply $\frac{8}{\sqrt{73}}$ by $\frac{\sqrt{73}}{\sqrt{73}}$ .

$\cos (θ) = \frac{8}{\sqrt{73}} \cdot \frac{\sqrt{73}}{\sqrt{73}}$

Combine and simplify the denominator.

Multiply $\frac{8}{\sqrt{73}}$ by $\frac{\sqrt{73}}{\sqrt{73}}$ .

$\cos (θ) = \frac{8 \sqrt{73}}{\sqrt{73} \sqrt{73}}$

Raise $\sqrt{73}$ to the power of $1$ .

$\cos (θ) = \frac{8 \sqrt{73}}{\sqrt{73} \sqrt{73}}$

Raise $\sqrt{73}$ to the power of $1$ .

$\cos (θ) = \frac{8 \sqrt{73}}{\sqrt{73} \sqrt{73}}$

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

$\cos (θ) = \frac{8 \sqrt{73}}{{\sqrt{73}}^{1 + 1}}$

Add $1$ and $1$ .

$\cos (θ) = \frac{8 \sqrt{73}}{{\sqrt{73}}^{2}}$

Rewrite ${\sqrt{73}}^{2}$ as $73$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{73}$ as $73^{\frac{1}{2}}$ .

$\cos (θ) = \frac{8 \sqrt{73}}{{(73^{\frac{1}{2}})}^{2}}$

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

$\cos (θ) = \frac{8 \sqrt{73}}{73^{\frac{1}{2} \cdot 2}}$

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

$\cos (θ) = \frac{8 \sqrt{73}}{73^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\cos (θ) = \frac{8 \sqrt{73}}{73^{\frac{2}{2}}}$

Rewrite the expression.

$\cos (θ) = \frac{8 \sqrt{73}}{73}$

Evaluate the exponent.

$\cos (θ) = \frac{8 \sqrt{73}}{73}$

Approximate the result.

$\cos (θ) = \frac{8 \sqrt{73}}{73} \approx 0.93632917$

Find the Cosine Given the Point (8,3)

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