The definition of the dot product was used in Section 1.9 to provide a straightforward derivation of the Law of Cosines. A similar argument can be used to derive the addition formulas for trigonometric functions, starting with cosine.

Consider the unit vectors $$\uu$$ and $$\vv\text{,}$$ shown on the unit circle in Figure 1.14. Their components are given by the coordinates of their endpoints on the unit circle, so elementary circle trigonometry implies that

\begin{align} \uu \amp= \cos\alpha\,\xhat + \sin\alpha\,\yhat ,\tag{1.10.1}\\ \vv \amp= \cos\beta\,\yhat + \sin\beta\,\yhat .\tag{1.10.2} \end{align}

Computing $$\uu\cdot\vv$$ algebraically, we obtain

$$\uu\cdot\vv = \cos\alpha\,\cos\beta + \sin\alpha\,\sin\beta .\tag{1.10.3}$$

On the other hand, since the angle $$\theta$$ between $$\uu$$ and $$\vv$$ is clearly $$\beta-\alpha\text{,}$$ and since $$\uu$$ and $$\vv$$ are unit vectors, the geometric formula for the dot product yields

$$\uu\cdot\vv = |\uu|\,|\vv|\,\cos\theta = \cos(\alpha-\beta) .\tag{1.10.4}$$

Equating these two expressions for the dot product yields

$$\cos(\alpha-\beta) = \cos\alpha\,\cos\beta - \sin\alpha\,\sin\beta\tag{1.10.5}$$

which is the addition formula for cosine.

Addition formulas for the remaining trigonometric functions can be found as usual, using relations such as $$\sin\theta=\cos(\frac\pi2-\theta)$$ and $$\tan\theta=\frac{\sin\theta}{\cos\theta}\text{.}$$