Exact value of cos(-π/8) , and sin(7π/8)?

Question

To find exact value of :

cos(-π/8)


sin(7π/8)

Answer 1

Use the half angle formula:

cos x/2 = sqrt((1 + cos x)/2). Set x = pi / 4

cos -pi/8 = cos pi/8 = sqrt((1 + sqrt(2)/2)/2) = sqrt(2 + sqrt(2))/2.

The half angle formula for sine is:

sin x/2 = sqrt((1 - cos x)/2). Again set x = pi / 4

sin (7pi / 8) = sin (pi/8) = sqrt(( 1 - sqrt(2)/2)/2) = sqrt(2 - sqrt(2))/2

Answer 2

Really putting us to the test, guy.
For the first one, you can invoke 2 sin x cos x = sin 2x ( I believe this is the sin double angle eq), noting that 2x = -pi/4 for which sin2 x = -sqrt(2)/2.
You can express sin x in terms of cos x using the sin^2+cos^2=1 relation, and then compute the answer.

For part 2, you can do the same routine where sin 2x is sin(7 pi/4)