Cool trick to solve questions in polar form

Many students are stuck when they see something like \frac{1}{1-e^{\frac{i\theta}{3}}} for example. They are unsure what tto do and some of them attempt to rationalise it.

Here’s a little tip to resolving all such questions. 🙂

Given 1-e^{n\theta} for n can be anything, fractional or integer, we simply factorise e^{\frac{n\theta}{2}} from the expression. Notice that 1=e^{i0} so using simple indices, this gives


Our objective of doing this is to have (e^{\frac{-n\theta}{2}}-e^{\frac{n\theta}{2}}). Now notice that this is a very familiar form, its z^{*}-z form which gives us -2iy=-2isin{\theta} for this case! Isn’t that convenient!






After some manipulation, we find the following result. In the event, have a fraction like one above, we still start with the same method. Hope this helps!

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