Determine the exact value of cos(Pi/24)???

Question

Determine the exact value of cos(Pi/24)???

Answer 1

cos(∏/24)=cos(7.5º)

first find cos(15º)

cos(15º)=cos(45º-30º)

use the identity cos(x-y)=cosxcosy+sinxsiny

cos45cos30+sin45sin30=

(√2/2)(√3/2)+(√2/2)(1/2)=

√6/4+√2/4=

(√6+√2)/4

cos(15º)=(√6+√2)/4

now use the half-angle formula cos(15º/2) because we know cos15º

cos(x/2)=√([1+cosx]/2)

√([1+cos15]/2)=

√([1+(√6+√2)/4]/2)=

√([4+√6+√2]/4)/2=

√([4+√6+√2]/8)=

√(4+√6+√2)/√8=

√(4+√6+√2)/√8=

√[(4+√6+√2)/2√2] is the exact value of cos(7.5º)

Also, I suppose you can rationalize the denominator if you like.

 

 

Answer 2

π/24 = 7.5°

 

First, find cos15° using the fact that 15° = 45° - 30°. then use a half angle identity to find cos7.5°.

 

cos15° = cos(45° - 30°)

 

           = cos45°cos30° + sin45°sin30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6+√2)/4

 

So, cos(π/24) = cos(7.5°) = √[(1+cos15°)/2]

 

                                       = √[(1+(√6+√2)/4)/2] = √[(4+√6+√2)/8]