Question

Hi, is there any specific formula or series for tan(nx) in terms of n and tan(x)?

Answer 1

salam

Answer 2

salam!

Answer 3

ye negahi be in bendaz
http://mathhelpforum.com/trigonometry/177739-tan-x.html

Answer 4

aaaali bud! mamnoon!

Answer 5

xahesh mikonam...baram jalebe age beshe formoole kolli bara n be dast ovord...

Answer 6

hatman roosh kar mikonam

Answer 7

nikasi thadi nake? putisi khudi khudi chinka phate pe ..

Answer 8

lol

Answer 9

kitcha "lol" phika jhinka jhinka tooon..

Answer 10

@ParsaN Salam,
zahesh mikonam Ka Sep... @mukushla "mathematical formula".
zey.

Answer 11

translate.google.com worked @ParthKohli :P

Answer 12

:D

Answer 13

@mathslover What's that? That's not the Irani I am learning at school...

Answer 14

that makes sense parth :D

Answer 15

:D

Answer 16

\[\cos nx+i \sin nx=e^{inx}=(e^{ix})^n=(\cos x+i \sin x)^n\\=\sum_{k=0}^{n} \left(\begin{matrix}n \\ k\end{matrix}\right)i^k \ \sin^k x \ \cos^{n-k}x=\cos^n x\sum_{k=0}^{n} \left(\begin{matrix}n \\ k\end{matrix}\right)i^k \ \sin^k x \ \cos^{-k}x\\=\cos^n x\sum_{k=0}^{n} \left(\begin{matrix}n \\ k\end{matrix}\right)i^k \ \tan^k x \]so we can say

\[\tan nx=\frac{Im\left( \sum_{k=0}^{n} \left(\begin{matrix}n \\ k\end{matrix}\right)i^k \ \tan^k x \right)}{Re\left( \sum_{k=0}^{n} \left(\begin{matrix}n \\ k\end{matrix}\right)i^k \ \tan^k x \right)}\]

Answer 17

and finally\[\tan nx=\frac{n\tan x-\left(\begin{matrix}n \\ 3\end{matrix}\right)\tan^3 x+\left(\begin{matrix}n \\ 5\end{matrix}\right)\tan^5 x+...}{1-\left(\begin{matrix}n \\ 2\end{matrix}\right)\tan^2 x+\left(\begin{matrix}n \\ 4\end{matrix}\right)\tan^4 x+...}\]

Answer 18

agha ino man takmil kardam..khedmate shoma..\[\large \tan nx=\frac{ \sum_{k=0}^{\lfloor \frac{n-1}{2} \rfloor} \left(\begin{matrix}n \\ 2k+1\end{matrix}\right) \ \tan^{2k+1} x}{\sum_{k=0}^{\lfloor \frac{n}{2} \rfloor} \left(\begin{matrix}n \\ 2k\end{matrix}\right) \ \tan^{2k} x }\]

Answer 19

fantastic! thanks a lot @mukushla ...

Answer 20

:)