Explanation of Sine Function Please

[ChrisAndre]

Lately I have been trying to plot sine waves, but I can't understand any explanation on the net. I always get y=sine(x), but I never receive any explanation regarding what x IS, be it angle or just the x-axis number. I found a way around this using the Pythagorean theorem, but that is annoying because I have to have a calculator to do it, and my goal is to be able to rattle off points with just brain power. Can someone help?

 

[fss]

x refers to the position on the x-axis. You may have seen y = sin (theta) with theta on the horizontal axis. In both cases you're usually talking about radians, so y = sin x crosses the horizontal axis at x = n*pi, n = ...-2, -1, 0, 1, 2...

 

[FlexGunship]

The ability to rattle of values of the sine function would be very impressive. I know the critical points (crossings and local minima and maxima), but certainly not the whole thing.

Asking for a way to memorize sine values is not unlike asking for a way to memorize pi. You just gotta' sit down and start memorizing. Fortunately, sine repeats... pi doesn't.

 

[JonF]

Consider the sine function as a function! Remember, functions are a rule where, it assigns each input to exactly one output. The sine function’s input is an angle (for the sake of plotting we’ll say it’s an angle in radians) the output is the ratio between the opposite side and the hypotenuse of the given angel.

So with this understanding if f(x) = sin(x), the “x” would be your input angel in radians. sin(x) would spit out whatever that ratio is.

 

[ChrisAndre]

I don't think I am getting this. If I have a circle with a radius of 10mm then hypotenuse is 10mm and the peak of a sine wave is 10mm. So far we have the first half of the sine function: sine(X)=opp/10mm. Now I need opposite, but that is dictated by radians. This is where I am stuck. How do I find opposite(y value) based on the radians?

I am pretty sure I understand it now. Tell me if this is right(I did not use the actual function, I assumed opp and adj were both 1, which spit out 45* which was converted to radians:

sin(0.785398163) = 1/1 = 1 = 45*, and therefore point 0.785398163 amplitude will be 1.

 

[JonF]

I'm confused as to what exactly you're asking.

Are you asking how to manually calculate the exact value of sin(x) for any given x? Or asking for a general understanding of what sin(x) is.

To derive the exact value of sin you need to take a calc course. But we can give you the formula, which is Sin(x) = Σ₀^∞ (-1)^n/(2n+1)!*x^(2n+1). If you want to get close you can take this sum to as many terms as desired instead of ∞.

 

[symbolipoint]

chrisAndre,
When you study Trigonometry, you make use of the unit circle. The sine function takes as input the angle rotation in radians of a ray of length 1 with endpoint at the origin, and the output is the coordinate of the horizontal axis. If you are studying Trigonometry, you very soon learn this. If you are not yet studying Trigonometry, then just refer to the unit circle in a Geometry book or a Trigonometry book.