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Math Help - Degree or Radian?

  1. #1
    Junior Member cupid's Avatar
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    Exclamation Degree or Radian?

    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) ?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    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.
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  3. #3
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    I believe you want radians.
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  4. #4
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    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.)
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  5. #5
    Junior Member cupid's Avatar
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    Quote Originally Posted by HallsofIvy View Post
    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
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  6. #6
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    Quote Originally Posted by cupid View Post
    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.
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  7. #7
    Junior Member cupid's Avatar
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    Quote Originally Posted by Plato View Post
    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] ???
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  8. #8
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    Quote Originally Posted by cupid View Post
    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.
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  9. #9
    Junior Member cupid's Avatar
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    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) ???
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  10. #10
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    Quote Originally Posted by cupid View Post
    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.
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  11. #11
    Junior Member cupid's Avatar
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    Quote Originally Posted by Plato View Post
    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
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  12. #12
    Behold, the power of SARDINES!
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    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
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  13. #13
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    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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