Thread: Question on complex no 2

Another question, I'm really sorry.

2.) Let z=cis(a)
a.) Use De Moivre's theorem to show that z^n + (1/z)^n = 2cos(na)
I was able to prove this part.
b.) Use the binomial theorem to expand (z + (1/z))^4
I got z^4 + 4z^2 +6 +(4/z^2) + (1/z^4), is it right?
Part 2, Hence show that cos^4(a) = (1/8) (cos(4a) + 4cos(2a) + 3 and find the integral of cos^4(a)da.
I'm not sure how to show..


Thank you so so much!
J.

Hey Tutu.

The expansion is correct.

For the last part consider the fact that e^(a*i*theta) = cos(a*theta) + i*sin(a*theta) = (e^(i*theta))^a = (cos(theta) + i*sin(theta)^a.

Thank you!

But how can cos^4(a) be substituted in e^(a*i*theta) ?
And I cannot sub it into cos(a*theta) + i*sin(a*theta) either because I will have the extra isin(a)..
I tried it out in another way, is it that
z^4 + 4z^2 +6 +(4/z^2) + (1/z^4) = 2cos(4a)?

Consider that cos is the real part and sin is the imaginary part.