1. For the complex number z = -4i:
(a) Find z1/5 (that is, all fifth roots of z). Leave your answer(s) in exponential form.

(b) Plot your answers from (a) onto a single Argand diagram. Label the axes, the solutions
to (a) themselves, and indicate the angles for each of the solutions.

(c) What relationship exists between the angles of successive fifth roots of z? That is, between
the angles of z1 and z2, the angles of z2 and z3, etc.

2. This question involves the identity (a statement that is true for all values of the variable, which here is ɵ)
cos(3 ɵ) = cos(ɵ)3 - 3 cos(ɵ) sin2(ɵ)

(a) State de Moivre's theorem.

(b) Expand (cos(ɵ) + i sin(ɵ))3 by multiplying out the brackets.

(c) Now use de Moivre's theorem to rewrite (cos(ɵ) + i sin(ɵ))3 in terms of cos(3 ɵ) and
sin(3 ɵ).

(d) Equate your two representations of (cos(ɵ) + i sin(ɵ))3, given by your results to part (b)
and part (c).

(e) Recall that if two complex numbers are equal, their real parts must be equal and their
imaginary parts must be equal. Use this fact, and your result from part (d), to confirm that the
identity given by equation (1) is correct.

(f) Use your result from part (d) to write an identity that expresses sin(3 ɵ) in terms of cos(ɵ)
and sin(ɵ).

please give me the detail solutions

Originally Posted by tutuxich

1. For the complex number z = -4i:
(a) Find z1/5 (that is, all fifth roots of z). Leave your answer(s) in exponential form.

This is not a homework service. You can pay to have these done. We are here to help you learn. And that is free.

There are five fifth roots of $-4i$

One of them is $\sigma=\sqrt[5]{4}\left[ {\cos \left( {\frac{{ - \pi }}{{10}}} \right) + i\sin \left( {\frac{{ - \pi }}{{10}}} \right)} \right]$.

Now let $\delta = \left[ {\cos \left( {\frac{{2\pi }}{5}} \right) + i\sin \left( {\frac{{2\pi }}{5}} \right)} \right]$.

The other four fifth roots are $\sigma\cdot\delta^k,~k=1,2,3,4$.

Originally Posted by tutuxich
(d) Equate your two representations of (cos(ɵ) + i sin(ɵ))3, given by your results to part (b)
and part (c).
This is a university math assignment that i am doing also. I don't need answers but would like to know what "equate" means?

Any help is appreciated.

Darman2142