[JCH2CNEFTG_20150410] Exponential Half-Power Trick

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 Euler’s Formula, one of the most 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^if  with its reciprocal  e^-if (which is also its 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^{2kq i},  force out the factor  e^{kq i}.  This gives you either a sin or cos expression.  For example, e^{6q i} – 1 = e^{3q i} (e^{3q i} – e^{-3q i}) = i×2e^{3q 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