Question

How do I get -(1+sqrt(3))/(2 sqrt(2)) from cos(11/12 pi) using half/double angle identities?

Answer 1

\[\cos(\frac{11}{12}\pi) =>-\frac{1+\sqrt{3}}{2\sqrt{2}}\]

Answer 2

well look at it like this

\[\cos(\frac{11\pi}{12}) = \cos(\frac{2\pi}{3} + \frac{\pi}{4})\]

Answer 3

then you need

\[\cos(A +B) = \cos(A)\cos(B) - \sin(A)\sin(B)\]

Answer 4

the sum of both angles are exact values....

just remember

\[\frac{2\pi}{3} ~is~2nd~quadrant\]

Answer 5

Well, ok. I was hoping not to have to use sum/difference formulas because they are longer than:\[\cos(\frac{\theta}{2})=\sqrt{\frac{1+\cos(\theta)}{2}}\]
But I guess I just have to not be lazy and evaluate more trig functions.

Answer 6

sum and difference are the easiest for this question

\[\cos(\frac{\pi}{4}) = \sin(\frac{\pi}{4}) = \frac{1}{\sqrt{2}}\]

\[\cos(\frac{2\pi}{3}) = - \frac{1}{2}~~~~\sin(\frac{2\pi}{3}) = \frac{\sqrt{3}}{2}\]

now just multiply the fractions... with is certainly easier than the half angle formula or double angle formula

Answer 7

That makes sense; a little bit of thinking ahead goes a long way.