삼각함수 공식 (Trigonometric Relations)

Trigonometric Relations  

1. Sum or Difference:
$ sin\left ( x\pm y \right )=sinx cosy\pm cosxsiny \\ cos\left ( x\pm y \right )=cosx cosy\mp sinxsiny \\ tan\left ( x\pm y \right )= \frac{tanx \pm tany}{1 \mp tanxtany} \\ sin^{2}x + cos^{2}x = 1 \\ tan^{2}x + 1 = sec^{2}x  \\ 1 + cot^{2}x = csc^{2}x $

2. Products into Sum or Difference:
$ 2sinxcosy = sin \left (x+y \right ) + sin \left (x-y \right ) \\ 2sinxsiny = -cos \left (x+y \right ) + cos \left (x-y \right ) \\ 2cosxcosy = cos \left (x+y \right ) + cos \left (x-y \right ) $

3. Sum or Difference into Products:
$ sinx+siny = 2sin\left (\frac{x+y}{2} \right )cos\left (\frac{x-y}{2} \right ) \\ sinx-siny = 2cos\left (\frac{x+y}{2} \right )sin\left (\frac{x-y}{2} \right ) \\ cosx-cosy = -2sin\left (\frac{x+y}{2} \right )sin\left (\frac{x-y}{2} \right ) \\ cosx+cosy = 2cos\left (\frac{x+y}{2} \right )cos\left (\frac{x-y}{2} \right ) $

4. Double and Half-angles:
$ sin2x = 2sinxcosx \\ cos2x = cos^{2}x - sin^{2}x = 1 - 2sin^{2}x = 2cos^{2}x - 1 \\ tan2x = \frac{2tanx}{1-tan^{2}x}\\ tan  \frac{x}{2} = \pm\sqrt{\frac{1-cosx}{1+cosx}}= \frac {sinx}{1+cosx}=\frac{1-cosx}{sinx} $

5. Series:

$\mathrm{sin}x=\frac{e^{jx}-e^{jx}}{2j}=x- \frac{x^3}{3!}+ \frac{x^5}{5!}- \frac{x^7}{7!}+... \\ \mathrm{cos}x=\frac{e^{jx}+e^{jx}}{2}=x- \frac{x^2}{2!}+ \frac{x^4}{4!}- \frac{x^6}{6!}+... \\ \mathrm{tan}x=\frac{e^{jx}-e^{jx}}{j\left ( e^{jx}+e^{jx} \right )}=x+ \frac{x^3}{3}+ \frac{2x^5}{15}+ \frac{17x^7}{315}+... $

6. Etc.:

$ \mathrm{sin}x = \mathrm{cos}\left ( x-\frac{\pi}{2} \right ) = \mathrm{cos}\left ( \frac{\pi}{2} -x\right ) \\ \mathrm{cos}x = \mathrm{sin}\left ( x+\frac{\pi}{2} \right ) = \mathrm{sin}\left ( \frac{\pi}{2} -x\right ) \\ \mathrm{sin}\left ( \pi - x \right )=\mathrm{sin}x \\ \mathrm{cos}\left ( \pi - x \right )=-\mathrm{cos}x \\
\mathrm{cot}x = \mathrm{cot}\left (  \pi +x  \right )= \mathrm{cot}\left (  2\pi +x  \right ) = -\mathrm{cot}\left (  \pi -x  \right )= -\mathrm{cot}\left (  2\pi -x  \right )$

 

	0°	30° (= 𝜋/6)	45° (= 𝜋/4)	60° (= 𝜋/3)	90° (= 𝜋/2)
sin	0	1/2	$ \sqrt{2}/2 $	$ \sqrt{3}/2 $	1
cos	1	$ \sqrt{3}/2 $	$ \sqrt{2}/2 $	1/2	0
tan	0	$ 1/ \sqrt{3}$	1	$ \sqrt{3} $	∞