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Please help me with these two questions

1. For the complex number z = -4i:
(a) Find z^1/5 (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).

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