Math Cribsheet

Addition Formulas:

(1)
\begin{eqnarray} \sin(\alpha-\beta)=\sin\alpha\cos\beta-\cos\alpha\sin\beta \\ \cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta \\ \sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta \\ \cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta \\ \tan(\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta} \\ \tan(\alpha-\beta)=\frac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta} \\ \end{eqnarray}

Law of Cosines:

(2)
\begin{align} c^2=a^2+b^2-2ab\cos\theta \end{align}

Law of Sines:

(3)
\begin{align} \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} \end{align}

Double Angle Formulas:

(4)
\begin{eqnarray} \sin2\alpha=2\sin\alpha\cos\alpha \\ \cos2\alpha=\cos^2\alpha-\sin^2\alpha \\ =2\cos^2\alpha-1 \\ =1-2\sin^2\alpha \end{eqnarray}

Half Angle Formulas:

(5)
\begin{eqnarray} \cos^2\frac{\alpha}{2}=\frac{1+\cos\alpha}{2} \\ \sin^2\frac{\alpha}{2}=\frac{1-\cos\alpha}{2} \\ \end{eqnarray}

Product Formulas:

(6)
\begin{eqnarray} \sin\alpha\cos\beta=\frac{1}{2}[\sin(\alpha-\beta)+\sin(\alpha+\beta)] \\ \sin\alpha\sin\beta=\frac{1}{2}[\cos(\alpha-\beta)-\cos(\alpha+\beta)] \\ \cos\alpha\cos\beta=\frac{1}{2}[\cos(\alpha-\beta)+\cos(\alpha+\beta)] \\ \end{eqnarray}

Miscellaneous

(7)
\begin{eqnarray} \sin^2\alpha+\cos^2\alpha=1 \\ \cos\left(\frac{\pi}{2}-\alpha\right)=\sin\alpha \\ \sin\left(\frac{\pi}{2}-\alpha\right)=\cos\alpha \\ \tan^2\theta+1=\sec^2\theta \\ 1+\cot^2\theta=\csc^2\theta \\ \end{eqnarray}
Bibliography
1. Howard Anton, Calculus.
3. Joseph W. Goodman, Introduction to Fourier Optics.
4. Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions
5. Nicholas George, Optical Systems (yellow book OPT 461)
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