A Level Further Mathematics for AQA Student Book 2 WORKED EXAMPLE 1.12 2πi
Let w = e 5 . a Show that Re(w) + Re(w 2) = − 1 . 2 b Hence find the exact value of cos 2π . 5 a From the diagram: Im
ω
1
The five points form a regular pentagon.
ω2
Re
1
ω3 ω4
–1
pl e
O
–1
w = w and w = (w 2) * Hence Re(w 4) = Re(w) and Re(w 3) = Re(w 2) 4
*
3
Using the result 1 + w + w + w + w = 0 and taking the real part: 1 + Re(w) + Re(w 2) + Re(w 3) + Re(w 4) = 0 3
4
Sa m
2
1 + 2Re(w) + 2Re(w 2) = 0
You are interested in the real parts. This is the result from Key point 1.9.
Pair up the terms with equal real parts.
Re(w) + Re(w ) = − 1 2 2
b Re(w) = cos 2π , Re(w 2) = cos 4π 5 5 cos 2π + cos 4π = −1 5 2 5 cos 2π + 2 cos2 2π − 1 = −1 5 2 5
4 cos2 2π + 2 cos 2π − 1 = 0 5 5
−2 + √4 + 16 cos 2π = 5 8 =
18
2πi Use the fact that w = e 5 = cos 2π + i sin 2π 5 5 and w 2 = cos 4π + i sin 4π . 5 5
Use cos 4π = 2 cos2 2π − 1. 5 5 This is a quadratic equation in cos 2π . 5 Take the positive root since cos 2π > 0. 5
−1 + √5 4
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