Trignometry Complex Number Assignment

Question 1 Let n be a positive integer of at least 2. Then,

l. for even n, sinn x may be expressed as the sum of cos(kv),

2. for odd n, Sinn x may be expressed as the sum of sin(kv), and

3. for any n, cosn x may be expressed as the sum of cos(kv) where k is some of {l, 2, n}.

(a) Verify the three statements above for n = 3, 4, 5 and 6. Use — + e-ix) and sin X = cosx — 2i

(b) Use the relevant result from (a) and trigonometric identitities to calculate the exact value of sin6 x given that cos(2x) = 2.

Question 2 Find the fourth roots of —8 — 8vfii. State them in polar form. Finally, sketch all on one complex plane.