Question

Find the exact value by using a half-angle identity.
cos(5pi/12)

Answer 1

\[\sin \frac{\theta}{2}=\sqrt{\frac{1-\cos \theta}{2}}\]

Answer 2

Use 5 pi/6 for theta

Answer 3

Why 5 pi/6? Why not 5pi/12?

Answer 4

Can someone do this for me step by step.

Answer 5

the half-angle identity presupposes that the angle is say \(\cfrac{\theta}{2}\) so in order to get say \(\theta\) then \(\theta = 2*\theta\) and \(
2*\cfrac{5\pi}{2}=\cfrac{5\pi}{6} \implies \cfrac{\cfrac{5\pi}{6}}{2} \implies \cfrac{\theta}{2}
\)

Answer 6

in order to get the angle as expected by the half-angle formula, the original/given angle must be DOUBLED, so once divided by 2, it returns the origina/given one :)

Answer 7

\(
2*\cfrac{5\pi}{12}=\cfrac{5\pi}{6} \implies \cfrac{\cfrac{5\pi}{6}}{2} \implies \cfrac{\theta}{2}
\)

Answer 8

So what is the exact value?

Answer 9

use the half-angle formula and your Unit Circle :)

Answer 10

sin ((5pi/12)/2) = Square root of 1-cos(5pi/12)/2 ?

Answer 11

$$
\cfrac{\cfrac{5\pi}{12}}{2} \implies \cfrac{5\pi}{24} \ne \cfrac{5\pi}{12}
$$

Answer 12

would the exact value be 1?

Answer 13

well, how did you get 1?

Answer 14

5pi/6 is 30 degrees which is 1/2 and 2 times 1/2 is 1?

Answer 15

well, the doubling is for the angle to be used inside the half-angle formula, not for any result to be doubled

Answer 16

So it is just 1/2?

Answer 17

$$
2*\cfrac{5\pi}{12}=\cfrac{5\pi}{6} \implies \cfrac{\cfrac{5\pi}{6}}{2} \implies \cfrac{\theta}{2} \implies cos\pmatrix{\cfrac{\theta}{2}} \implies cos\pmatrix{\cfrac{\boldsymbol{\cfrac{5\pi}{6}}}{2}}\\
cos\pmatrix{\cfrac{\boldsymbol{\cfrac{5\pi}{6}}}{2}}=\sqrt{\cfrac{1+cos\pmatrix{\cfrac{5\pi}{6}}}{2}}
$$