**Question**

**Introduction**

This is a complex-number
question that appeared for ‘A’ Level Mathematics in November 1998. Remember that mathematics is never
out-dated. Many Singapore schools keep this sort of
questions in their question banks (for tutorial exercises, tests and
examinations), in the hope that it becomes part of students’ repertoire.

Students are
expected to know the famous

**, one of the most***Euler’s Formula**beautiful*formulas discovered by this visually-challenged but prolific mathematician. It links the exponential function with trigonometry via the idea of angle rotation. Adding e^{i}*with its*^{f}*reciprocal*e^{-i}*(which is also its*^{f}*conjugate*) gives a cosine expression, while subtracting gives a sine expression.
The
above question can be solved by rationalising the denominator and using heavy trigonometry
and half-angle formulas. There is
nothing wrong with this approach. I am
going to illustrate a kewl approach, using what I call the exponential “half-power trick”.
Basically, whenever you see an expression like 1 ± e

^{2k}^{q}^{ i}, force out the factor e^{k}^{q}^{ i}. This gives you either a sin or cos expression. For example, e^{6}^{q}^{ i}– 1 = e^{3}^{q}^{ i}(e^{3}^{q}^{ i}– e^{-3}^{q}^{ i}) = i×2e^{3}^{q}^{ i}sin*q*.**Solution**

Observe that we have killed two birds with one stone. At the last step, we just compare the real and
imaginary parts to read off the answers.

**Suitable Levels**

* GCE ‘A’ Level H2
Mathematics (“Complex Numbers”)

* precocious students who
love complex numbers

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