Question

If cos(20) - sin(20) = p, find the expression for cos(40) in terms of p.

Answer 1

cos(40) = cos^2 (20) - sin^2 (20) = (cos20 - sin20)(cos20 + sin20) was my first thought.

Answer 2

yes, but (cos20 + sin20) is going to unknown.

Answer 3

You are thinking good though :)

Answer 4

That's what the question is all about!

Answer 5

Well, I guess you don't need more help with this one.

Answer 6

I do :(

Answer 7

@ganeshie8 @Kainui

Answer 8

Oh nvm... sin(A/2) + cos(A/2) = sqrt(1 + sinA)

Answer 9

cos40 = (cos20 - sin20)(cos20 + sin20)

what i'm suggesting is
cos20 + sin20 = cos20 + cos70 = 2cos45cos(-25) = (cos25)sqrt2 --- 1

cos20 - sin20 =p (given)
sin70 - sin20 = p
2sin45cos25=p
(cos25)sqrt2 =p --- 2

using 1 and 2 ; cos20 + sin20 = p

cos40 = p^2

Answer 10

^^^^ If cos(20) - sin(20) = p can cos(20) + sin(20) also be p?

Answer 11

\[
\cos(20) - \sin(20) = p \\
\text{Square both sides:} \\
\cos^2(20) + \sin^2(20) - 2\cos(20)\sin(20) = p^2 \\
1 - 2\cos(20)\sin(20) = p^2 \\
2\cos(20)\sin(20) = 1 - p^2\\
(\cos(20)+\sin(20))^2 = \cos^2(20) + \sin^2(20) + 2\cos(20)\sin(20) = 1 + 1 - p^2 \\
(\cos(20)+\sin(20)) = \sqrt{2-p^2} \\
\text{ } \\
\cos(40) = \cos^2 (20) - \sin^2 (20) = (\cos(20) - \sin(20))(\cos(20) + \sin(20))
= p\sqrt{2-p^2} \\
\]