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Math Help - What IS sin?

  1. #1
    Senior Member Paze's Avatar
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    What IS sin?

    So Sin is the y-axis on the unit circle. Why? It can't be the length of the opposite side since it can be negative values.

    Sin is also defined as a function. Why? To me it seems that sin is a fixed number depending on the placement of the y-coordinate on the unit circle. So what is it that sin does to a number for it to be defined as a function? How does sin convert 30 into 0.5?

    Thanks!
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  2. #2
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    Re: What IS sin?

    Consider the unit circle. Start at the point (1,0). Move up along the circle an arc length of X - then sin(X) will be your y-coordinate and cos(X) will be your x-coordinate. For example, the diameter of this circle is 2, so the circumference is 2*Pi: this is why the period if 2*Pi. This is only true in radians though, which is why they are the natural way to define the functions.
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  3. #3
    Senior Member Paze's Avatar
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    Re: What IS sin?

    Quote Originally Posted by SworD View Post
    Consider the unit circle. Start at the point (1,0). Move up along the circle an arc length of X - then sin(X) will be your y-coordinate and cos(X) will be your x-coordinate. For example, the diameter of this circle is 2, so the circumference is 2*Pi: this is why the period if 2*Pi. This is only true in radians though, which is why they are the natural way to define the functions.
    I know. I am asking why that is? Thank you though.
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  4. #4
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    Re: What IS sin?

    Because that is the definition! If you want to understand the sin function, don't start from the triangle definition, its harder to work with. Use what I said, and then notice that in the 1st quadrant (we can assume without loss of generality that the angle will be in this quadrant, because no angle in a right triangle is greater than 90 degrees), it does seem to be equivalent, because the hypotenuse is 1, so "opposite/hypotenuse" is just the y-coordinate divided by 1. But in order to generalize it to hypotenuses other than 1, it follows that you need to divide the y-coordinate by the radius so keep the proportion the same. Someone can probably explain this better.

    But its a function because it takes a number and gives another. Suppose you wanted to find sin(5). You would walk 5 units around the arc of the circle, and then your y-coordinate is the value of the function.

    Think of it like this: the triangle SOHCAHTOA thing is just one particular use, out of many, of the trigonometric functions.
    Last edited by SworD; April 1st 2013 at 12:36 AM.
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  5. #5
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    Re: What IS sin?

    In formula form the definition of sine is

    sin(x)= \frac{e^{ix}-e^{-ix}}{2i}
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    Re: What IS sin?

    "Sin" is breaking one of the commandments!

    "Sine", on the other hand, is a trig function. Whatever definition of "sine" you use you can quickly get " sin^2(x)+ cos^2(x)= 1". That, in particular, means that the graph of the relation given by the parametric equations, x= cos(t), y= sin(t) is the unit circle x^2+ y^2= 1. If you interpret "t" to be the distance around the circumference of the unit circle to a given point, starting from (1, 0), and so the angle a line through the origin and that point makes with the x-axis, measured in radians, then, simply because "y= sin(t)" "sine is the y-axis". That is based on a number of conventions. If we were to take parametric equations x= sin(t), y= cos(t) we would have "sine" as the x-coordinate, not the y. And if we started measuring the angle somewhere other than with the x-axis we would get quite a different result.
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  7. #7
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    Re: What IS sin?

    Geometric definition of sin of a:

    Angle a: Arc length divided by radius
    Sin(a): y coordinate of point on unit circle at angle a.

    Mathematics definition: Sum of an infinite series. Also from infinite series definition of e^(ia) via post #5.
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  8. #8
    Senior Member Paze's Avatar
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    Re: What IS sin?

    Thanks guys. This has given me some insight into Sine (and taught me how to spell it). I will now do some more homework to fully understand it!
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