How do you find the exact value of cos( 2 arctan (5/12) )cos(2arctan(512))?

1 Answer
Jun 8, 2016

119/169119169

Explanation:

Use the cosine double angle formula:

cos(2x)=2cos^2(x)-1cos(2x)=2cos2(x)1

Here, inside the cosine function, the angle arctan(5/12)arctan(512) is being doubled, so we see that

cos(2arctan(5/12))=2cos^2(arctan(5/12))-1cos(2arctan(512))=2cos2(arctan(512))1

To find cos(arctan(5/12))cos(arctan(512)), draw a picture where tan(beta)=5/12tan(β)=512, or where 55 is the opposite side's length and 1212 is the adjacent side's length.

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In this triangle, 1313 is the hypotenuse (through the Pythagorean Theorem), and cos(beta)=12/13cos(β)=1213. This also means that cos(arctan(5/12))=12/13cos(arctan(512))=1213.

So we see that

2cos^2(arctan(5/12))-1=2(12/13)^2-1=288/169-1=119/1692cos2(arctan(512))1=2(1213)21=2881691=119169