14.7 Cool stuff with trigonometry
14.7.1 Proving identities
We can write as the real part of . This because is equal to the real part of , which is equal to .
Express in terms of .
Solution: Firstly, we can write as the equation
We can then apply De Moivre’s theorem55 5 to rewrite the expression in terms of and
We can now expand the binomial obtained, which leads to the result that
Then, we can tidy this up a bit, leading to the expression
We are only interested in the imaginary parts of the expansion, so it is therefore equal to just
We want in terms of , however! There’s a rogue gatecrasher 66 6 A handy way to remember whether or shows a certain property is (as previously mentioned, TODO: mention) that generally behaves ”better” than . in the previous expression - the ! Fortunately we can remove the without too much difficulty using the Pythagorean identity.
Thus, we can say that
14.7.2 Writing complex numbers in terms of the exponential function
Using Euler’s formula, it is possible to write both and in terms of . As , and we can either add or subtract these two quantities in order to write both trigonometric functions in terms of .
For , we can add and .
Thus we can say that
for all values of x. 77 7 Which looks remarkably like a hyperbolic function!.
We can do a similar thing for .
Which means that
14.7.3 Using the exponential form to show odd/evenness
The cosine function is even, that is
|As addition is commutative||(14.73)|
The sine function is odd, that is