De Moivre’s theorem

In this question you must show detailed reasoning.

Use de Moivre’s Theorem to show that, if cos5θ ≠ 0,

By considering the equation tan5 θ = 1, use the result in part (a) to find the exact roots of the equation

Give the roots in the form t = tan φ where 0 < φ < π.

By first expressing t⁴ - 4t³ - 14t² - 4t + 1 = 0 in the form (t - 1)⁴ = kt², where k is a constant to be determined, show that

In this question you must show detailed reasoning.

Use de Moivre’s theorem to determine constants A, B and C such that

sin⁴θ ≡ A cos 4θ + B cos 2θ + C.

The function f is defined by

The diagram shows the curve with equation y = for 0 ⩽ x < 1 and the asymptote x = 1. The region R is the unbounded region between the curve, the x-axis, the line x = 0 and the line x = 1.

You are given that the area of R is finite.

Determine the exact area of R.

Use de Moivre’s theorem to find the constants A, B and C in the identity

sin⁵θ ≡ A sin θ + B sin 3θ + C sin 5θ.

You are given that the cubic equation 2x³ + px² + qx - 3 = 0, where p and q are real numbers, has a

Write down a second complex root, β.

Determine the third root, γ.

Find the value of p and the value of q.