Deriving the sin and cos addition formulas using Euler's formula

Question

I am trying to figure out the quick way to remember the addition formulas for $\sin$ and $\cos$ using Euler's formula:

$$\begin{align} \sin(\alpha + \beta) &= \sin \alpha\;\cos\beta + \cos\alpha\;\sin\beta \\ \cos(\alpha + \beta) &= \cos \alpha\;\cos\beta - \sin\alpha\;\sin\beta \\ \sin(\alpha-\beta) &= \sin\alpha\;\cos\beta - \cos\alpha\;\sin\beta \\ \cos(\alpha-\beta) &= \cos\alpha\;\cos\beta + \sin\alpha\;\sin\beta \end{align}$$

I'm now convinced of why it is true that $e^{ix} = \cos(x) + i\sin(x)$ but I don't know how to use this to derive these four rules.

Calculate $e^{i(\alpha+\beta)}$ and $e^{i(\alpha-\beta)}$ using the fact that $e^{i(x+y)} = e^{ix}e^{iy}$ — Dionel Jaime, Feb 26 '18 at 03:29
If you're looking for a way to remember the identities, this answer of mine may help. — Blue, Feb 26 '18 at 03:39
@Blue I did see that post but it almost seems like a ton more to remember than just trying to brute memorize the formulas — user525966, Feb 26 '18 at 03:52

Harry Alli · Accepted Answer · 2018-02-26T03:34:45.470

2

$$e^{ix}=\cos x+i\sin x$$ $$Re\{e^{i(x+y)}\}=\cos(x+y)$$ $$=Re\{e^{ix}e^{iy}\}$$ $$=Re\{(\cos x+i\sin x)(\cos y+i\sin y)\}$$ $$=Re\{\cos x\cos y+i(\sin x\cos y+\sin y\cos x)-\sin y\sin x\}$$ $$=\cos x\cos y-\sin x\sin y$$

$$Im\{e^{i(x+y)}\}=\sin(x+y)=\sin x\cos y+\sin y\cos x$$

You can apply a similar reasoning to all the other ones, just working with the laws of exponents and comparing the real and imaginary parts.

As for why $e^{ix}=\cos x+i\sin x$ applies, this is commonly understood by considering the Taylor series expansions of $e^{ix}$ and comparing it with that of $\sin x$ and $\cos x$

edited Feb 26 '18 at 03:34

answered Feb 26 '18 at 03:28

Harry Alli

2,091

What about subtraction? Do you have to divide these? – user525966 Feb 26 '18 at 03:38
Nope, just write $\cos (-x)+i\sin(-x)=\cos x-i\sin x$ as cosine is even, sine is off. – Harry Alli Feb 26 '18 at 03:39
But the lefthand side is still $e^{ix} / e^{iy}$? – user525966 Feb 26 '18 at 03:41
$e^{i(x-y)}=e^{ix}\times e^{-iy}=(\cos x + i\sin x)(\cos y-i\sin y)$ – Harry Alli Feb 26 '18 at 03:42
Can the tangent addition/subtraction formulas be extracted from this somehow too or is that a separate thing? – user525966 Feb 26 '18 at 04:00

Deriving the sin and cos addition formulas using Euler's formula

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