More trigonometric formulas

Sum-to-product formulas. By letting $ u = \dfrac{\alpha + \beta}{2}$ and $ v = \dfrac{\alpha - \beta}{2}$ in (3-1)-(3-4), we can verify the following formulas:

\begin{displaymath}\begin{array}{ll}
\sin\alpha + \sin\beta =
2 \sin\dfrac{\alph...
...c{\alpha + \beta}{2} \sin\dfrac{\alpha - \beta}{2}.
\end{array}\end{displaymath}

Sum of trigonometric functions having common period. Let $ \theta = \tan^{-1}(b/a)$ if $ a \ge 0$ (we set $ \theta = \pi + \tan^{-1}(b/a)$ if $a < 0$). Then we have $ \cos\theta = \dfrac{a}{\sqrt{a^2 + b^2}}$ and $ \sin\theta = \dfrac{b}{\sqrt{a^2 + b^2}}$, and can verify

$\displaystyle a \cos\omega x + b \sin\omega x
= \sqrt{a^2 + b^2}[\cos\omega x \cos\theta
+ \sin\omega x \sin\theta]
= \sqrt{a^2 + b^2}\cos(\omega x - \theta).
$



Department of Mathematics
Last modified: 2005-09-29