Cosine and Sine Angle Addition Intuition

Question

I am lacking in understanding in the cosine and sine angle addition formulas. I have seen several questions similar to this but I have not seen an answer that explains how this conclusion can be derived. Geometric proofs offer little intuition and the rotation matrix is derived from the cosine and sin angle addition formulas. I haven't been able to derive this from calculus either and am already struggling on figuring out the sine and cosine derivatives in a way that makes sense instead of blindly accepting it. I would appreciate it if someone could answer why the cosine and sine angle addition formulas are as they are in an intuitive manner.

Thanks, Jackson

Answer 1

You can. I approach trig via the unit circle. Here is the fundamental theorem of trig.

Theorem. Let $\theta\in\mathbb{R}$, $(x,y)\in\mathbb{R}^2$. Then, if you rotate the vector $xi + yj$ through angle $\theta$, the coordinates of the "head" of this vector will be $$(x\cos(\theta) - y \sin(\theta)), x\sin(\theta) + y\cos(\theta)).$$

To prove this, you should project the rotated vector onto $(x,y)$ and $(y, -x)$. You can do this via some simple triangle similarity arguments.

Once you have this, the addition formulae for sin and cos fall right out. See if you can think this through.

Answer 2

For example you want to prove:$$\cos (\alpha+\beta) = \cos (\alpha)\cos (\beta) - \sin (\alpha)\sin (\beta)$$.

You can use the identities: $$e^{i(\alpha+\beta)} = e^{i\alpha}\cdot e^{i\beta}$$,and

$$e^{i\alpha} = \cos (\alpha) + i\sin(\alpha)$$

Answer 3

One of the cleanest derivations is based on Euler's formula which tells us that $e^{i \theta} = \cos \theta + i \sin \theta$. The derivation works by noting that

\begin{align} \cos(x + y) + i \sin(x + y) & = e^{i (x + y)} \\ {} & = e^{ix} e^{iy} \\ {} & = (\cos x + i \sin x)(\cos y + i \sin y) \\ \end{align}

and expanding the product on the last line. I'll leave the details for you to work out.