Question 14 - A-Level Maths

Exam Board	OCR MEI
Module	Further Pure Core (Further Pure Core)
Session	Specimen
Topic	Complex numbers 2

Starting with the result $$\mathrm { e } ^ { \mathrm { i } \theta } = \cos \theta + \mathrm { i } \sin \theta$$ show that
(A) $( \cos \theta + \mathrm { i } \sin \theta ) ^ { n } = \cos n \theta + \mathrm { i } \sin n \theta$
(B) $\cos \theta = \frac { 1 } { 2 } \left( \mathrm { e } ^ { \mathrm { i } \theta } + \mathrm { e } ^ { - \mathrm { i } \theta } \right)$.
Using the result in part (i) (A), obtain the values of the constants $a , b , c$ and $d$ in the identity
Using the result in part (i) (B), obtain the values of the constants $P , Q , R$ and $S$ in the identity
Show that $\cos \frac { \pi } { 12 } = \left( \frac { 26 + 15 \sqrt { 3 } } { 64 } \right) ^ { \frac { 1 } { 6 } }$.
Using the result in part (i) (A), obtain the values of the constants $a , b , c$ and $d$ in the identity $$\cos 6 \theta \equiv a \cos ^ { 6 } \theta + b \cos ^ { 4 } \theta + c \cos ^ { 2 } \theta + d$$ $$\cos ^ { 6 } \theta \equiv P \cos 6 \theta + Q \cos 4 \theta + R \cos 2 \theta + S$$