cant figure out the exact value

Question

1)  Determine the exact value of cos 7.5°.

Answer 1

cos 30 degrees = square root of 3 divided by 2

use 1/2 angle formula to find cos15

when you get that, use the half angle formula to get cos7.5

Answer 2

find exact value of cos7.5°

first find cos(45°-30°) because 45-30=15 (and then we'll work with 15/2=7.5)

cos(15°)=cos(45°-30°)=cos45cos30+sin45sin30)

cos45cos30+sin45sin30=(√2/2)(√3/2)+(√2/2)(1/2)=

(√6/4)+(√2/4)=

(√6+√2)/4=cos(15°)

now find cos(7.5°)=cos(15°/2)

cos(15°/2)=√((1+cos15°)/2)=

√{(1+[√6+√2]/4)}/2=

√{(4+√6+√2)/4}/2=

√{(4+√6+√2)/8}

(1/2)√{(4+√6+√2)/2} is the exact value

check: 2√{(4+2.449489+1.414213)/2}=

(1/2)√{(7.863702)/2}=

(1/2)√3.93185=

(1/2)*1.982889=

0.991444, and...

cos7.5°=0.99144486 (if we used more digits in √6 and √2 we would have gotten the ......86