Law of Cosines, Trigonometric Angle Addition Theorems, and Dot Product Relations Just as the derivative, slope, and gradient are essentially the same thing I've realized that the Law of Cosines, trigonometric angle addition, and dot product are saying the same thing. My question is why is the trigonometric angle addition theorems the same thing as the dot product? Also could the trigonometric angle addition theorems be proved using the derivatives of sine and cosine?
Thanks,
Jackson
 A: Use the following based on the definition of $cos(x,y)$:
$cos(x,y) = \frac{\left \langle x,y \right \rangle}{\sqrt{\left \langle x,x \right \rangle\left \langle y,y \right \rangle}}$
So in any unitary space, consider the following facts:


*

*It is not necessarily real and therefore not necessarily the cosine of the angle. 

*It is not symmetric because $cos(x,y) \neq {cos(y,x)}$
Therefore:


*

*$\left | cos(x,y) \right |\leq 1$ 

*$cos(y,x)= $ the complex conjugate of $cos(x,y)$ 

*$cos(\alpha x,x) = \frac{\alpha }{\left | \alpha  \right |}$ for  $\alpha \in \mathbb{C} $
now we can begin to dissect the problem restricting outsells to Euclidean space.
Now say angle(x,y) = $\left | cos^{-1}\right |$ on the interval $ \left [ 0, \frac{\pi }{2} \right ]$
The real part of cos(x,y) plays the part of of cos in a unitary law of cosines.
Only the real part of $cos(x,y)$ is involved in a unitary law of cosines.
${\left \| x+y \right \|}^{2}$ = $\left \| x \right \|^{2} + \left \| y \right \|^{2} + 2 \left \| x \right \|\left | y \right |\Re cos(x,y)$
Now, if you take $\mathbb{C}^{n}$ and apply the "standard" complex inner product, the real part of this inner product is just that "standard" inner product on $\mathbb{R}^{2n}$.
The imaginary part is the sum of the (oriented) areas of the parallelograms formed by the corresponding pairs of complex numbers. That gives some intuition, perhaps, for the geometry of the "complex part" of the unitary dot product.
