How do you find the exact value of $cos (2 arctan (\frac{5}{12}))$ $cos( 2 arctan (5/12) )$ ?

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How do you find the exact value of $cos (2 arctan (\frac{5}{12}))$ $cos( 2 arctan (5/12) )$ ?

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Answer 1 · 2016-06-08T04:09:43

$\frac{119}{169}$ $119/169$

Explanation:

Use the cosine double angle formula:

$cos (2 x) = 2 {cos}^{2} (x) - 1$ $cos(2x)=2cos^2(x)-1$

Here, inside the cosine function, the angle $arctan (\frac{5}{12})$ $arctan(5/12)$ is being doubled, so we see that

$cos (2 arctan (\frac{5}{12})) = 2 {cos}^{2} (arctan (\frac{5}{12})) - 1$ $cos(2arctan(5/12))=2cos^2(arctan(5/12))-1$

To find $cos (arctan (\frac{5}{12}))$ $cos(arctan(5/12))$ , draw a picture where $tan (β) = \frac{5}{12}$ $tan(beta)=5/12$ , or where $5$ $5$ is the opposite side's length and $12$ $12$ is the adjacent side's length.

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In this triangle, $13$ $13$ is the hypotenuse (through the Pythagorean Theorem), and $cos (β) = \frac{12}{13}$ $cos(beta)=12/13$ . This also means that $cos (arctan (\frac{5}{12})) = \frac{12}{13}$ $cos(arctan(5/12))=12/13$ .

So we see that

$2 {cos}^{2} (arctan (\frac{5}{12})) - 1 = 2 {(\frac{12}{13})}^{2} - 1 = \frac{288}{169} - 1 = \frac{119}{169}$ $2cos^2(arctan(5/12))-1=2(12/13)^2-1=288/169-1=119/169$

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How do you find the exact value of $cos (2 arctan (\frac{5}{12}))$ $cos( 2 arctan (5/12) )$ ?

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Explanation:

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