Trigonometry/More About Addition Formulas

Related Formulae

\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 Formulae

\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 $0^{\circ }\leq a\leq 90^{\circ }$ then the trigonometric functions are all positive so the positive sign is needed before the square root.

Derivations

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

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 Formulae

\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}}}

Derivations

$\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)$
$\cos(a-b)=\cos(a)\cos(b)+\sin(a)\sin(b)$

Using $\cos(a+b)$ and the fact that cosine is even and sine is odd, we have

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