Trigonometry/More About Addition Formulas

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Related FormulaeEdit

\sin(a-b)=\sin(a)\cos(b)-\cos(a)\sin(b)
\sin(2a)=2\sin(a)\cos(a)
\sin\bigl(\tfrac{a}{2}\bigr)=\pm\sqrt{\frac{1-\cos(a)}{2}}

Tangent FormulaeEdit

\tan(a+b)=\frac{\tan(a)+\tan(b)}{1-\tan(a)\tan(b)}
\tan(a-b)=\frac{\tan(a)-\tan(b)}{1+\tan(a)\tan(b)}
\tan(2a)=\frac{2\tan(a)}{1-\tan^2(a)}=\frac{2\cot(a)}{\cot^2(a)-1}=\frac{2}{\cot(a)-\tan(a)}
\tan\bigl(\tfrac{a}{2}\bigr)=\pm\sqrt{\frac{1-\cos(a)}{1+\cos(a)}}=\frac{\sin(a)}{1+\cos(a)}=\frac{1-\cos(a)}{\sin(a)}=\frac{-1\pm\sqrt{1+\tan^2(a)}}{\tan(a)}

In the last row of expressions, if 0^\circ\le a\le 90^\circ then the trigonometric functions are all positive so the positive sign is needed before the square root.

DerivationsEdit

  • \sin(a+b)=\sin(a)\cos(b)+\cos(a)\sin(b)

Using cofunctions we know that \sin(a)=\cos(90^\circ-a) . Use the formula for \cos(a-b) and cofunctions we can write

\sin(a+b) =\cos(90-(a+b))
=\cos\bigl((90^\circ-a)-b\bigr)
=\cos(90^\circ-a)\cos(b)+\sin(90^\circ-a)\sin(b)
={\color{red}\sin(a)\cos(b)+\cos(a)\sin(b)}


  • \sin(a-b)=\sin(a)\cos(b)-\cos(a)\sin(b)

Having derived \sin(a+b) we replace b with -b and use the fact that cosine is even and sine is odd.

\sin\bigl(a+(-b)\bigr) =\sin(a)\cos(-b)+\cos(a)\sin(-b)
=\sin(a)\cos(b)+\cos(a)\bigl(-\sin(b)\bigr)
={\color{red}\sin(a)\cos(b)-\cos(a)\sin(b)}

Related FormulaeEdit

\cos(a-b)=\cos(a)\cos(b)+\sin(a)\sin(b)
\cos(2a)=\cos^2(a)-\sin^2(a)=2\cos^2(a)-1=1-2\sin^2(a)
\cos\bigl(\tfrac{a}{2}\bigr)=\pm\sqrt{\frac{1+\cos(a)}{2}}

DerivationsEdit