Trigonometric Relation Formulas

[TheDestroyer]

I know some of them, such as :

cos (2x) = (cos x)^2 - (sin x)^2

sin (2x) = 2(cos x)(sin x)

tan (2x) = (2tan (x))/(1-(tan x)^2)

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

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

I need the other formulas such as

sin x/2
cos x/2
sin x^2
cos x^2
sin x^3
cos x^3
sin (x^(1/2))
cos (x^(1/2))

And any others, everyyything about them,

Anyone can help? or guide me to a link?

[PrudensOptimus]

Tan[x +- y] = \frac{Tan(x) +- Tan (y)}{1 -+ Tan(x)Tan(y)}

Sin^{2}(x) + Cos^{2}(x) = 1

\frac{Sin A}{A} = \frac{Sin B}{B}

A^{2} = B^{2} + C^{2} - 2BC Cos A

Cos A = \frac{A^{2} - B^{2} - C^{2}}{-2BC}

Sin(\frac{x}{2}) = +-\sqrt{\frac{1 - Cos(x)}{2}}

Cos(\frac{x}{2}) = +-\sqrt{\frac{1 + Cos(x)}{2}}

Sin^{2}(\frac{x}{2}) = \frac{1 - Cos(x)}{2}

Cos^{2}(\frac{x}{2}) = \frac{1 + Cos(x)}{2}

Sin^{2}(x) = \frac{1 - Cos(2x)}{2}

Cos^{2}(x) = \frac{1 + Cos(2x)}{2}

[TheDestroyer]

Thank you, but I still need sin (x^1/2), cos (x^1/2), sin (x^2), cos (x^2), sin (x^3), cos (x^3), sin (x^1/3), cos (x^1/3)

The roots and poweres are for the angles not for the function

:)

[PrudensOptimus]

Sin[sqrt(x)] ? ... i'm gonna leave that for the expert of this forum to answer lol. But I will cogitate on it.


Edit: After playing around with a right triangle a bit, I got this from a right triangle with sides a,b,c, with c being the hyp, and a being the opp side with angle x.


sin(x) = \frac{a}{c}

x = sin^{-1}\frac{a}{c}

\sqrt(x) = \sqrt{sin^{-1}\frac{a}{c}} = \phi


\begin{equation*}
\begin{split}
sin \phi = +- \sqrt{\frac{1 - cos(2\phi)}{2}}
&= +- \sqrt{\frac{1 - cos(2\sqrt{sin^{-1}(\frac{a}{c})})}{2}}
\end{split}
\end{equation*}

[TheDestroyer]

Thank you for making a try, anyone else can help also?

[gnome]

google search "trigonometric identities" will give you thousands of sites

[PrudensOptimus]

Originally posted by gnome
google search "trigonometric identities" will give you thousands of sites


but not the ones he want.

[chroot]

How about

http://functions.wolfram.com

It's the base of relationships used by the Mathematica software and has every identity, I believe, known to man.

- Warren

[master_coda]

I doubt you'll find any identities for those functions. At least nothing that isn't even messier than the original function. I can think of some ugly expansions, like:


\sin\left(x^2\right)=\sum_{n=0}^\infty(-1)^n\frac{x^{2+4n}}{(2n+1)!}


Of course, I could be wrong. I don't know how to prove that there isn't a simple identity.

[StephenPrivitera]

how about sin(arctanx)? can this be simplified?
how about sin(arccosx)?
I can get this:
sinx=cos(pi/2-x)
y=sinx=cos(pi/2-x)
x=arcsiny=pi/2-arccosy
so arccosy=pi/2-arcsiny
sin(arccosy)=sin(pi/2-arcsiny)=cos(arcsiny)

[master_coda]

\begin{align*}
\sin(\arctan(x))&=\frac{x}{\sqrt{1+x^2}} \\
\sin(\arccos(x))&=\sqrt{1-x^2}
\end{align*}

[master_coda]

Now that I think about it we also have:


\begin{align*}
\cos(\arcsin(x))&=\sqrt{1-x^2} \\
\cos(\arctan(x))&=\frac{1}{\sqrt{1+x^2}} \\
\tan(\arcsin(x))&=\frac{x}{\sqrt{1-x^2}} \\
\tan(\arccos(x))&=\frac{\sqrt{1-x^2}}{x}
\end{align*}


Using those identities, I can spot a few more identities, like


\sin(\arccos(x))=\cos(\arcsin(x))


which was already mentioned, as well as


\begin{align*}
\sin(\arctan(x))&=x\cos(\arctan(x)) \\
\tan(\arcsin(x))&=\frac{1}{\tan(\arccos(x))}
\end{align*}

[PrudensOptimus]

Originally posted by master_coda

\begin{align*}
\sin(\arctan(x))&=\frac{x}{\sqrt{1+x^2}} \\
\sin(\arccos(x))&=\sqrt{1-x^2}
\end{align*}


= dSin^{-1}(x)/dx

[master_coda]

I thought


\frac{d}{dx}(\arcsin(x))=\frac{1}{\sqrt{1-x^2}}