Section 3.10 Addition Formulas
The definition of the dot product was used in Section 3.9 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 3.10.1. 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{3.10.1}\\
\vv \amp= \cos\beta\,\yhat + \sin\beta\,\yhat .\tag{3.10.2}
\end{align}
Computing \(\uu\cdot\vv\) algebraically, we obtain
\begin{equation}
\uu\cdot\vv = \cos\alpha\,\cos\beta + \sin\alpha\,\sin\beta .\tag{3.10.3}
\end{equation}
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
\begin{equation}
\uu\cdot\vv = |\uu|\,|\vv|\,\cos\theta = \cos(\alpha-\beta) .\tag{3.10.4}
\end{equation}
Equating these two expressions for the dot product yields
\begin{equation}
\cos(\alpha-\beta) = \cos\alpha\,\cos\beta - \sin\alpha\,\sin\beta\tag{3.10.5}
\end{equation}
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 \(\tan\theta=\frac{\sin\theta}{\cos\theta}\text{.}\)
