Appendix E Some Trigonometry
\begin{align*}
\sin^2 x + \cos^2 x \amp= 1\\
1 + \tan^2 x \amp= \sec^2 x\\
\cos^2(ax) \amp= \frac12[1 + \cos(2ax)]\\
\sin^2(ax) \amp= \frac12[1 - \cos(2ax)]\\
\sin(ax) \cos(ax) \amp= \frac12 \sin(2ax)\\
\sin(x \pm y) \amp= \sin x \cos y \pm \cos x \sin y\\
\cos(x \pm y) \amp= \cos x \cos y \mp \sin x \sin y\\
\cos x \cos y \amp= \frac12 \left[ \cos(x-y) + \cos(x+y) \right]\\
\sin x \sin y \amp= \frac12 \left[ \cos(x-y) - \cos(x+y) \right]\\
\sin x \cos y \amp= \frac12 \left[ \sin(x-y) + \sin(x+y) \right]
\end{align*}
Values at key angles.
In the first two quadrants, the main values with simple forms are
\begin{equation*}
\begin{array}[t]{|c||>{\ds}c|>{\ds}c|>{\ds}c|>{\ds}c|>{\ds}c||>{\ds}c|>{\ds}c|>{\ds}c|>{\ds}c|}
\hline
\theta \amp 0 (\rightarrow) \amp \frac{\pi}{6} \amp \frac{\pi}{4} \; (\nearrow) \amp \frac{\pi}{3} \amp \frac{\pi}{2} \, (\uparrow)\amp \frac{2\pi}{3} \amp \frac{3\pi}{4} (\nwarrow) \amp \frac{5\pi}{6} \amp \pi (\leftarrow)
\\[1ex] \hline\hline
\sin\theta \amp 0 \amp \frac{1}{2} \amp \frac{1}{\sqrt{2}} \amp \frac{\sqrt{3}}{2} \amp 1 \amp \frac{\sqrt{3}}{2} \amp \frac{1}{\sqrt{2}} \amp \frac{1}{2} \amp 0
\\[1ex] \hline
\cos\theta \amp 1 \amp \frac{\sqrt{3}}{2} \amp \frac{1}{\sqrt{2}} \amp \frac{1}{2} \amp 0 \amp -\frac{1}{2} \amp -\frac{1}{\sqrt{2}} \amp -\frac{\sqrt{3}}{2} \amp -1
\\[1ex] \hline
\tan\theta \amp 0 \amp \frac{1}{\sqrt{3}} \amp 1 \amp \sqrt{3} \amp \mbox{DNE} \amp -\sqrt{3} \amp -1 \amp -\frac{1}{\sqrt{3}} \amp 0
\\[1ex] \hline
\end{array}
\end{equation*}
When sketching curves, it can help to know some numerical values:
\begin{equation*}
\sqrt{2} \approx 1.4142 \quad \text{and} \quad \sqrt{3} \approx 1.7321
\end{equation*}
leading to
\begin{equation*}
\frac{1}{\sqrt{2}} \approx 0.7071, \quad
\frac{1}{\sqrt{3}} \approx 0.5774, \quad \text{and} \quad
\frac{\sqrt{3}}{2} \approx 0.8660.
\end{equation*}
