Question

Find the value of cos theta if cos 5 theta = 0

Answer 1

Step-by-step explanation:

So here we will use De Moivre's theorem here:

As you stated we will use the fact that cos(5θ)+isin(5θ)=(cos(θ)+isin(θ))5cos⁡(5θ)+isin⁡(5θ)=(cos⁡(θ)+isin⁡(θ))5, so get this just in terms of cos(5θ)cos⁡(5θ) we can write cos(5θ)=R(cos(θ)+isin(θ))5cos⁡(5θ)=ℜ(cos⁡(θ)+isin⁡(θ))5

So expanding this we have:

cos(5θ)=R(cos5θ+5icos4θsinθ+10i2cos3θsin2θ+10i3cos2θsin3θ+5i4cosθsin4θ+i5sin5θ)cos⁡(5θ)=ℜ(cos5⁡θ+5icos4⁡θsin⁡θ+10i2cos3⁡θsin2⁡θ+10i3cos2⁡θsin3⁡θ+5i4cos⁡θsin4⁡θ+i5sin5⁡θ)

Since the real parts cannot have an odd power of sinθsin⁡θ (since it will not give a real number) we get :

cos(5θ)=cos5θ+10