Find the Cosine (-2,pi/6)

$(- 2, \frac{π}{6})$

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- 2, \frac{π}{6})$ , draw the triangle between the three points $(0, 0)$ , $(- 2, 0)$ , and $(- 2, \frac{π}{6})$ .

Opposite : $\frac{π}{6}$

Adjacent : $- 2$

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

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

$\sqrt{4 + {(\frac{π}{6})}^{2}}$

Apply the product rule to $\frac{π}{6}$ .

$\sqrt{4 + \frac{π^{2}}{6^{2}}}$

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

$\sqrt{4 + \frac{π^{2}}{36}}$

To write $4$ as a fraction with a common denominator, multiply by $\frac{36}{36}$ .

$\sqrt{4 \cdot \frac{36}{36} + \frac{π^{2}}{36}}$

Combine $4$ and $\frac{36}{36}$ .

$\sqrt{\frac{4 \cdot 36}{36} + \frac{π^{2}}{36}}$

Simplify the expression.

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Combine the numerators over the common denominator.

$\sqrt{\frac{4 \cdot 36 + π^{2}}{36}}$

Multiply $4$ by $36$ .

$\sqrt{\frac{144 + π^{2}}{36}}$

Rewrite $\sqrt{\frac{144 + π^{2}}{36}}$ as $\frac{\sqrt{144 + π^{2}}}{\sqrt{36}}$ .

$\frac{\sqrt{144 + π^{2}}}{\sqrt{36}}$

Simplify the denominator.

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

$\frac{\sqrt{144 + π^{2}}}{\sqrt{6^{2}}}$

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

$\frac{\sqrt{144 + π^{2}}}{6}$

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{- 2}{\frac{\sqrt{144 + π^{2}}}{6}}$ .

$\frac{- 2}{\frac{\sqrt{144 + π^{2}}}{6}}$

Simplify $\cos (θ)$ .

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Multiply the numerator by the reciprocal of the denominator.

$\cos (θ) = - 2 \frac{6}{\sqrt{144 + π^{2}}}$

Multiply $\frac{6}{\sqrt{144 + π^{2}}}$ by $\frac{\sqrt{144 + π^{2}}}{\sqrt{144 + π^{2}}}$ .

$\cos (θ) = - 2 (\frac{6}{\sqrt{144 + π^{2}}} \cdot \frac{\sqrt{144 + π^{2}}}{\sqrt{144 + π^{2}}})$

Combine and simplify the denominator.

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Multiply $\frac{6}{\sqrt{144 + π^{2}}}$ and $\frac{\sqrt{144 + π^{2}}}{\sqrt{144 + π^{2}}}$ .

$\cos (θ) = - 2 \frac{6 \sqrt{144 + π^{2}}}{\sqrt{144 + π^{2}} \sqrt{144 + π^{2}}}$

Raise $\sqrt{144 + π^{2}}$ to the power of $1$ .

$\cos (θ) = - 2 \frac{6 \sqrt{144 + π^{2}}}{\sqrt{144 + π^{2}} \sqrt{144 + π^{2}}}$

Raise $\sqrt{144 + π^{2}}$ to the power of $1$ .

$\cos (θ) = - 2 \frac{6 \sqrt{144 + π^{2}}}{\sqrt{144 + π^{2}} \sqrt{144 + π^{2}}}$

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

$\cos (θ) = - 2 \frac{6 \sqrt{144 + π^{2}}}{{\sqrt{144 + π^{2}}}^{1 + 1}}$

Add $1$ and $1$ .

$\cos (θ) = - 2 \frac{6 \sqrt{144 + π^{2}}}{{\sqrt{144 + π^{2}}}^{2}}$

Rewrite ${\sqrt{144 + π^{2}}}^{2}$ as $144 + π^{2}$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{144 + π^{2}}$ as ${(144 + π^{2})}^{\frac{1}{2}}$ .

$\cos (θ) = - 2 \frac{6 \sqrt{144 + π^{2}}}{{({(144 + π^{2})}^{\frac{1}{2}})}^{2}}$

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

$\cos (θ) = - 2 \frac{6 \sqrt{144 + π^{2}}}{{(144 + π^{2})}^{\frac{1}{2} \cdot 2}}$

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

$\cos (θ) = - 2 \frac{6 \sqrt{144 + π^{2}}}{{(144 + π^{2})}^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\cos (θ) = - 2 \frac{6 \sqrt{144 + π^{2}}}{{(144 + π^{2})}^{\frac{2}{2}}}$

Divide $1$ by $1$ .

$\cos (θ) = - 2 \frac{6 \sqrt{144 + π^{2}}}{144 + π^{2}}$

Simplify.

$\cos (θ) = - 2 \frac{6 \sqrt{144 + π^{2}}}{144 + π^{2}}$

Multiply $- 2 \frac{6 \sqrt{144 + π^{2}}}{144 + π^{2}}$ .

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Combine $- 2$ and $\frac{6 \sqrt{144 + π^{2}}}{144 + π^{2}}$ .

$\cos (θ) = \frac{- 2 (6 \sqrt{144 + π^{2}})}{144 + π^{2}}$

Multiply $6$ by $- 2$ .

$\cos (θ) = \frac{- 12 \sqrt{144 + π^{2}}}{144 + π^{2}}$

Move the negative in front of the fraction.

$\cos (θ) = - \frac{12 \sqrt{144 + π^{2}}}{144 + π^{2}}$

Approximate the result.

$\cos (θ) = - \frac{12 \sqrt{144 + π^{2}}}{144 + π^{2}} \approx - 0.9673972$

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