SOLUTION: Find the value of without using calculator: sin^4 ( π/8)+sin^4 (3π/8)+sin^4 (5π/8)+sin^4 (7π/8)

Algebra ->  Trigonometry-basics -> SOLUTION: Find the value of without using calculator: sin^4 ( π/8)+sin^4 (3π/8)+sin^4 (5π/8)+sin^4 (7π/8)      Log On


   

Question 1023864: Find the value of without using calculator: sin^4 ( π/8)+sin^4 (3π/8)+sin^4 (5π/8)+sin^4 (7π/8)
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
sin%5E4+%28pi%2F8%29%2Bsin%5E4+%283pi%2F8%29%2Bsin%5E4+%285pi%2F8%29%2Bsin%5E4+%287pi%2F8%29
=2sin%5E4+%28pi%2F8%29%2B2sin%5E4+%283pi%2F8%29,
because sin+%28pi%2F8%29+=+sin+%287pi%2F8%29 and sin%283pi%2F8%29+=+sin+%285pi%2F8%29, since sin%28pi-x%29+=+sinx
Now because of the fact that sinx+=+cos%28pi%2F2-x%29, the last expression is equal to
<----Equation (A)
Now cos2x+=+2cos%5E2%28x%29-1 implies that
cos%282%2A%28pi%2F8%29%29+=+cos%28pi%2F4%29+=+sqrt%282%29%2F2+=+2cos%5E2%28pi%2F8%29-1, hence
cos%5E2%28pi%2F8%29+=+%28sqrt%282%29+%2B2%29%2F4
==> cos%5E4%28pi%2F8%29+=+%283%2B2sqrt%282%29%29%2F8
Thus,

after substitution into Equation A.