AQA
Further Paper 2
2023
June
— Question 15
5 marks
Exam Board
AQA
Module
Further Paper 2 (Further Paper 2)
Year
2023
Session
June
Marks
5
Topic
Complex numbers 2
15
Given that \(z = \cos \theta + \mathrm { i } \sin \theta\), use de Moivre's theorem to show that
$$z ^ { n } - z ^ { - n } = 2 i \sin n \theta$$
15
The series \(S\) is defined as
$$S = \sin \theta + \sin 3 \theta + \ldots + \sin ( 2 n - 1 ) \theta$$
Use part (a) to express \(S\) in the form
$$S = \frac { 1 } { 2 \mathrm { i } } \left( G _ { 1 } \right) - \frac { 1 } { 2 \mathrm { i } } \left( G _ { 2 } \right)$$
where each of \(G _ { 1 }\) and \(G _ { 2 }\) is a geometric series.