Degree or Radian?

**cupid** · March 4th 2011, 03:10 AM

suppose i have a function sin(sin x)
sin x will give values [-1,1] ...
so now, should i take these values in degree or radian to find sin(sin x) ?

**FernandoRevilla** · March 4th 2011, 03:41 AM

The standard "geometric" definition for:

$\sin:\mathbb{R}\to [-1,1],\quad x\to \sin x$

$\cos: \mathbb{R}\to [-1,1],\quad x\to \cos x$

is when $x$ is "measured" in radians. So we have good properties, for example:

(i) $(\sin)'=\cos$

(ii) $\sin x=x-x^3/3!+x^5/5!-\ldots\quad (\forall x\in \mathbb{R})$

etc.

**Perfessor** · March 4th 2011, 11:14 AM

I believe you want radians.

**HallsofIvy** · March 5th 2011, 01:19 AM

The only time you should use degrees is when the problem specifically gives angles in degrees. For problems in which sine (or cosine) is used just as a function always use radians.

(There are a number of ways to define sine and cosine that do not involve angles at all. So strictly speaking the argument is not in "radians" or "degrees". However, to be able to identify those functions with the sine and cosine functions we learned in trigonometry, we use radians.)

**cupid** · March 6th 2011, 04:12 AM

Originally Posted by HallsofIvy

The only time you should use degrees is when the problem specifically gives angles in degrees. For problems in which sine (or cosine) is used just as a function always use radians.

(There are a number of ways to define sine and cosine that do not involve angles at all. So strictly speaking the argument is not in "radians" or "degrees". However, to be able to identify those functions with the sine and cosine functions we learned in trigonometry, we use radians.)

Actually My question is that the output of sine and cosine functions is in degrees or radians

**Plato** · March 6th 2011, 04:29 AM

Originally Posted by cupid

Actually My question is that the output of sine and cosine functions is in degrees or radians

For every $x\in\mathbb{R}$ it is the case that $\sin(x)\in\mathbb{R}$ and $\cos(x)\in\mathbb{R}$ .
Does that answer the question.

**cupid** · March 6th 2011, 07:56 AM

Originally Posted by Plato

For every $x\in\mathbb{R}$ it is the case that $\sin(x)\in\mathbb{R}$ and $\cos(x)\in\mathbb{R}$ .
Does that answer the question.

Isn't it more like ...

$sinx \in [-1,1]$ ???

**Plato** · March 6th 2011, 08:02 AM

Originally Posted by cupid

Isn't it more like ... $sinx \in [-1,1]$ ???

But $[-1,1]\subset\mathbb{R}$ .

The point being that both the sine and cosine functions map real numbers to real numbers.

**cupid** · March 6th 2011, 08:10 AM

but my question is that ... for example i have a question ... sin (sin x)

i need to calculate the value of this function at $x = \frac{\pi}{2}$

as $sin\frac{\pi}{2} = 1$
now i need to calculate sin(1)

so should i find value of sin(1 radian) or sin(1 degree) ???

**Plato** · March 6th 2011, 08:19 AM

Originally Posted by cupid

but my question is that ... for example i have a question ... sin (sin x)
i need to calculate the value of this function at $x = \frac{\pi}{2}$
as $sin\frac{\pi}{2} = 1$ now i need to calculate sin(1)
so should i find value of sin(1 radian) or sin(1 degree) ???

I do not know what a degree is.
I know what real numbers are.
1 is a real number.
Thus it is $\sin\left(\sin\left(\frac{\pi}{2}\right)\right)=\s in(1)\approx 0.841$ .

**cupid** · March 6th 2011, 11:29 AM

Originally Posted by Plato

I do not know what a degree is.
I know what real numbers are.
1 is a real number.
Thus it is $\sin\left(\sin\left(\frac{\pi}{2}\right)\right)=\s in(1)\approx 0.841$ .

that is for 1 radian ... for 1 degree its like 0.0014 something

**TheEmptySet** · March 6th 2011, 11:48 AM

The point that I think you are missing is that Radians are a pure number! They are defined as the arc length of a circle divided by the radius of the same circle so strictly speaking the "units" (length) reduce out leaving a Real number.

Radian - Wikipedia, the free encyclopedia

**HallsofIvy** · March 6th 2011, 12:14 PM

Yes, that was what I was trying to say to begin with!

"Strictly speaking the argument is not in "radians" or "degrees". However, to be able to identify those functions with the sine and cosine functions we learned in trigonometry, we use radians."

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