Math (Trigonometry [Polar Form])
posted by Shin .
Let z be a complex number such that z = 2(cos 8∘ + i cos 82∘).Then z^5 can be expressed as r(sin α∘+ i cos α∘), where r is a real number and 0 ≤ α ≤ 90. What is the value of r+α?
Hint to solve:
Let z be a complex number such that z = 2(cos 5∘ + i cos 85∘). Then z^6 can be expressed as r(sin α∘ + i cos α∘), where r is a real number and 0 ≤ α ≤90. What is the value of r+α?
Applying De Moivre's formula gives
Since cos30∘=cos(90∘−60∘)=sin60∘ and sin30∘=sin(90∘−60∘)=cos60∘,
Therefore, r=64 and α=60, hence r + α = 64 + 60 = 124.