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Section 1.10 Addition Formulas

Figure 1.14. The derivation of the trigonometric addition formulas using the definition of the dot product.

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

\begin{equation} \uu\cdot\vv = \cos\alpha\,\cos\beta + \sin\alpha\,\sin\beta .\tag{1.10.3} \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{1.10.4} \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{1.10.5} \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 \(\tan\theta=\frac{\sin\theta}{\cos\theta}\text{.}\)