My favourite formula

November 8th, 2007 | Categories: general math | Tags:

Are you geeky enough to have a favourite formula? Because I am and it turns out that I am not the only one either. I have just googled ‘favourite formula’ and the second result was a link to the aply named ‘Mathematics Weblog‘ where the author described an equation usually known as Euler’s formula.

$\light e^{i \pi} = -1$

The Nobel prize winning physicist, Richard Feynman, once referred to this equation as ‘the most remarkable formula in mathematics’ and he really knew his stuff. Now I agree with Feynman, it really is an amazing formula since it connects some of the most important constants of mathematics in a small and elegant package, but it’s not my favourite. Elegant it may be but it’s not particularly useful.

My personal favourite is the general case of the above equation. Also called Euler’s equation, it is written as

$\light e^{ix}=\cos(x)+ i \sin(x)$

Set $\light x$ to $\light \pi$ and out pops Feynman’s favourite but we can do so much more with it than that and in this post I am going to focus on using it to obtain the trigonometric addition identities.

I have always struggled to remember trig identities – No matter how hard I try, I somehow manage to get the signs wrong but with a little bit of work I can just derive them using Euler’s equation and be sure that I have got them right. Start off by writing

$\light e^{i(a+b)} = e^{ia}e^{ib}$

now substitute Euler’s formula in both sides of the equation

$\light cos(a+b) + i sin(a+b) = (cos(a) + i sin(a)) (cos(b)+i sin(b))$

Multiplying out the right hand side leads to

$\light cos(a+b) + i sin(a+b) = cos(a) cos(b)-sin(a) sin(b) + i ( cos(a)sin(b) + sin(a)cos(b) )$

Equating real and imaginary parts on both sides gives us two trig identities for the price of one

$\light cos(a+b) = cos(a) cos(b)-sin(a) sin(b)$

$\light sin(a+b) = cos(a)sin(b) + sin(a)cos(b)$

We can get similar equations for $\light cos(a-b)$ and $\light sin(a-b)$ by simply substituting -b for b.

In my dim and distant past I used to be an assistant instructor for undergraduate physics problems classes and I found that many students had not seen this derivation before but they felt that it was obvious (and rather useful) once they had been shown it.

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