# Math Help - What IS sin?

1. ## 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!

2. ## 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.

3. ## Re: What IS sin?

Originally Posted by SworD
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.

4. ## 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.

5. ## Re: What IS sin?

In formula form the definition of sine is

$sin(x)= \frac{e^{ix}-e^{-ix}}{2i}$

6. ## 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.

7. ## 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.

8. ## 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!