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My confusion is how do we define : $\sin (x)$ for $x\in \mathbb{R}$.

I only know that $\sin(x)$ is defined for degrees and radians..

Suddenly, I have seen what is $\sin (2)$..

I have no idea how to interpret this when not much information is given what $2$ is...

does this mean $2$ radians or $2$ degrees or some thing else...

I always wanted to clarify this but could not do it...

I guess most of the school students have this confusion..

please help me to understand this...

Thank you....

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    Well, when you see the real function value $;\sin 2;$ you're usually using radians: that's the usual way to extend the definition of the trigonometric functions from a (the) straight angle triangle to the unit circle and to the whole real line. – DonAntonio Oct 22 '13 at 11:14
  • Usually, if it were degrees, they would've (should've) written $\sin (2^\circ)$ – Arthur Oct 22 '13 at 11:15

2 Answers2

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To completely specify the sine function, you must specify the unit of angular measure. It is most common in mathematical parlance to use radians. You are correct to be concerned about this ambiguity.

ncmathsadist
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  • Thank you for your clarification :) –  Oct 22 '13 at 11:24
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    Among other very important things, if the angle is not in radians then there are several things that do not work, for instance: the derivatives of sine or of cosine are not what we're used to...:) – DonAntonio Oct 22 '13 at 11:27
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You are using radians in your case. The most common definition of the sine is $\sin(x) := \sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n+1)!} x^{2n+1}$ though, which coincides with the sine in radians as you know it.

Arthur
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  • $\sin (x)$ is real number where as $\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n+1)!} x^{2n+1}$ should be radians.. I am a bit more confused when i see this... :O –  Oct 22 '13 at 11:23
  • It is common to define the sine as $\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n+1)!} x^{2n+1}$, which makes sense for every real number $x$. This definition of the sine gives the function that you know as the 'sine in radians'. – Arthur Oct 22 '13 at 11:25
  • It's not that that sum "should be radians", @PraphullaKoushik. What it actually means is that if you want to attach to that number $;x;$ a geometric meaning then it must be the measure of angle by the arc if circle it intersects. But for this, you should take $;x;$ as any other usual, real number....because that's what it is! – DonAntonio Oct 22 '13 at 11:26
  • @DonAntonio : It does strike something but i did not understand everything.. May be i need some time... –  Oct 22 '13 at 11:30