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

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


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

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


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

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


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