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Welcome to NRICH.

cos(7p/12) in surd form

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By Anonymous on Monday, March 12, 2001 - 04:14 pm:

How do you evaluate some thing like this in surd form? cos(7p/12) = ?

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By Michael Doré (Md285) on Monday, March 12, 2001 - 04:29 pm:

You can use the double angle formula. We have:

cos(7p/6) = 2cos²(7p/12) - 1

(using cos 2A = 2cos²A - 1)

The left hand side is -cos(p/6) = -sqrt(3)/2.

So

cos²(7p/12) = 1/2 - sqrt(3)/4

Therefore (as cos(7p/12 < 0) we see the answer is:

-sqrt(1/2 - sqrt(3)/4)

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By Kerwin Hui (Kwkh2) on Monday, March 12, 2001 - 04:33 pm:

Another way is to note that 7p/12=p/3+p/4. So

cos(7p/12)=cos(p/4) cos p/3 - sin(p/4) sin(p/3) = 1/sqrt(2)×1/2-1/sqrt(2)×sqrt(3)/2=2^-1.5(1-3 ^0.5)

Kerwin

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By Anonymous on Monday, March 12, 2001 - 04:55 pm:

Thank you Michael, Kerwin.

Michael, how did you figure out that cos(7p/12) < 0?

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By Barkley Bellinger (Bb246) on Monday, March 19, 2001 - 04:45 pm:

A previous anonymous questioner asked how Michael knew that cos(7p/12)<0. If you know your cosine graph then you will know that cos(x)<0 for p/2<x<p, which 7p/12 satisfies.

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