Algebra Examples

$(6, - 4)$

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(6, - 4)$ , draw the triangle between the three points $(0, 0)$ , $(6, 0)$ , and $(6, - 4)$ .

Opposite : $- 4$

Adjacent : $6$

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

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Raise $6$ to the power of $2$ .

$\sqrt{36 + {(- 4)}^{2}}$

Raise $- 4$ to the power of $2$ .

$\sqrt{36 + 16}$

Add $36$ and $16$ .

$\sqrt{52}$

Rewrite $52$ as $2^{2} \cdot 13$ .

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Factor $4$ out of $52$ .

$\sqrt{4 (13)}$

Rewrite $4$ as $2^{2}$ .

$\sqrt{2^{2} \cdot 13}$

Pull terms out from under the radical.

$2 \sqrt{13}$

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{6}{2 \sqrt{13}}$ .

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

Simplify $\cos (θ)$ .

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Cancel the common factor of $6$ and $2$ .

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Factor $2$ out of $6$ .

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

Cancel the common factors.

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Factor $2$ out of $2 \sqrt{13}$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\cos (θ) = \frac{3}{\sqrt{13}}$

Multiply $\frac{3}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

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

Combine and simplify the denominator.

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Multiply $\frac{3}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\cos (θ) = \frac{3 \sqrt{13}}{\sqrt{13} \sqrt{13}}$

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

$\cos (θ) = \frac{3 \sqrt{13}}{\sqrt{13} \sqrt{13}}$

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

$\cos (θ) = \frac{3 \sqrt{13}}{\sqrt{13} \sqrt{13}}$

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

$\cos (θ) = \frac{3 \sqrt{13}}{{\sqrt{13}}^{1 + 1}}$

Add $1$ and $1$ .

$\cos (θ) = \frac{3 \sqrt{13}}{{\sqrt{13}}^{2}}$

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

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

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

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

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

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

$\cos (θ) = \frac{3 \sqrt{13}}{13^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\cos (θ) = \frac{3 \sqrt{13}}{13^{\frac{2}{2}}}$

Rewrite the expression.

$\cos (θ) = \frac{3 \sqrt{13}}{13}$

Evaluate the exponent.

$\cos (θ) = \frac{3 \sqrt{13}}{13}$

Approximate the result.

$\cos (θ) = \frac{3 \sqrt{13}}{13} \approx 0.83205029$