De Moivre’s theorem

Use de Moivre’s theorem to show that

cos3θ = 4cos³θ – 3cosθ

Use de Moivre’s theorem to express sin3θ in terms of sinθ

Hence show that

Given that z cos i sin , use de Moivre’s theorem to show that

The series S is defined as

Use part (a) to express S in the form

where each of G₁ and G₂ is a geometric series.

Hence, show that

The Argand diagram shows the solutions to the equation z⁵ 1

Solve the equation

giving your answers in the form z cos i sin , where 0 2

Explain why the points on an Argand diagram which represent the solutions found in part (a) are the vertices of a regular pentagon.

Show that if c cos , where z cos i sin is a solution to the equation z⁵ 1, then c satisfies the equation

The Argand diagram on page 22 is repeated below.

Explain, with reference to the Argand diagram, why the expression

has a repeated quadratic factor.

Two of the solutions to the equation cos 6 0 are and

Find the other solutions to the equation cos 6 0 for 0

Use the fact that and are solutions to the equation cos 6 0 to find

a and b are integers.

Hence show that

If z cos i sin , use de Moivre’s theorem to prove that

Express sin⁵ in terms of sin 5, sin 3 and sin

Hence show that