FrankXie
Active Member
Re: HSC 2014 4U Marathon
for part (ii):
![](https://latex.codecogs.com/png.latex?\bg_white $Consider the geometric series of complex numbers$ \sum_{n=0}^\infty[(\cos\theta+i\sin\theta)\cos\theta]^n )
Because the modulus of common ratio is less than 1, the limiting sum exists and
![](https://latex.codecogs.com/png.latex?\bg_white S_\infty=\frac{1}{1-\cos\theta(\cos\theta+i\sin\theta)})
![](https://latex.codecogs.com/png.latex?\bg_white =\frac{1}{\sin\theta(\sin\theta-i\cos\theta)})
![](https://latex.codecogs.com/png.latex?\bg_white =\frac{\sin\theta+i\cos\theta}{\sin\theta})
Finally equate the real part of both sides.
for part (ii):
Because the modulus of common ratio is less than 1, the limiting sum exists and
Finally equate the real part of both sides.
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