14.5: Frequently Used Trigonometric and Calculus Expressions

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Appendix E.PNG

\begin{aligned} &\sin \theta=\mathrm{a / c} \\[4pt] &\cos \theta=\mathrm{b / c} \\[4pt] &\tan \theta=\mathrm{a / b} \\[4pt] &\mathrm{a^{2}+b^{2}=c^{2}}
\sin ^{2} \theta+\cos ^{2} \theta=1 \\[4pt]
&\mathrm{e}^{\mathrm{j} \theta}=\cos \theta+\mathrm{j} \sin \theta \\[4pt]
&(\mathrm{d} / \mathrm{d} \theta) \sin \theta=\cos \theta \\[4pt]
&(\mathrm{d} / \mathrm{d} \theta) \cos \theta=-\sin \theta \\[4pt]
&(\mathrm{d} / \mathrm{d} \mathrm{x}) \mathrm{e}^{\mathrm{f}(\mathrm{x})}=[\mathrm{df}(\mathrm{x}) / \mathrm{d} \mathrm{x}] \mathrm{e}^{\mathrm{f}(\mathrm{x})} \\[4pt]
&\mathrm{a}^{\mathrm{x}}=\left(\mathrm{e}^{\ln \mathrm{a}}\right)^{\mathrm{x}} \\[4pt]
&(\mathrm{d} / \mathrm{d} \mathrm{x}) \mathrm{x}^{\mathrm{n}}=\mathrm{n} \mathrm{x}^{\mathrm{n}-1} \\[4pt] &(\mathrm{d} / \mathrm{dx}) \mathrm{AB}=\mathrm{A}(\mathrm{dB} / \mathrm{dx})+\mathrm{B}(\mathrm{d} \mathrm{A} / \mathrm{dx}) \\[4pt] &(\mathrm{d} / \mathrm{dx}) \mathrm{f}_{1}\left[\mathrm{f}_{2}(\theta)\right]=\left[\mathrm{df}_{1} / \mathrm{d} f_{2}\right]\left[\mathrm{d} f_{2}(\theta) / \mathrm{d} \theta\right] \mathrm{d} \theta / \mathrm{d} \mathrm{x} \\[4pt] &(\mathrm{d} / \mathrm{dx}) \sin [\mathrm{f}(\theta)]=\cos [\mathrm{f}(\theta)][\mathrm{df}(\theta) / \mathrm{d} \theta] \mathrm{d} \theta / \mathrm{d} \mathrm{x} \\[4pt] &\int \sin \theta \mathrm{d} \theta=-\cos \theta \\[4pt] &\int \cos \theta \mathrm{d} \theta=\sin \theta \\[4pt] &\int \mathrm{e}^{\mathrm{ax}} \mathrm{dx}=\mathrm{e}^{\mathrm{ax}} / \mathrm{a} \\[4pt] &\int \mathrm{x}^{\mathrm{n}} \mathrm{dx}=\mathrm{x}^{\mathrm{n}+1} /(\mathrm{n}+1) \end{aligned}