Why Does the Sine Function Create a Squiggly Line on the Number Plane?

[aricho]

hi,

i was wondering why the sine function makes a squiggly line on the number plane. i know its not the right terminology so sorry for that.

thanks

 

[finchie_88]

If you drew 360 right angled triangles, increasing the angle by one degree each time, and measured the value of the opposite/hypotenuse wrt to the angle in question, then you will get a sine curve. The fact that it repeats is simply because geometrically speaking, 45 degrees is that same as 405 degrees etc.

 

[berkeman]

Here's a fun way to visualize what is going on with the sine wave curve (or cosine). Picture a wheel with a bright spot at one location on its circumference. Look at the wheel from the side, so that it looks like a circle to you. Now when you spin the wheel, the spot goes round and round, right? And if you imagine a horizontal x-axis and a vertical y-axis passing through the middle of the wheel, you can draw two right triangles at any instant from the middle of the wheel out to the spot. Those are the triangles that you use to calculate the angle of the radius to the spot, with respect to the x and y axes.

Now turn the wheel so that you see it edge on. Spin the wheel again so that you see the spot going up and down as the wheel spins. And finally, move the wheel from left to right at a constant velocity, and watch the shape of the curve that is traced out by the spot. It will be the sine wave curve that you are asking about. Pretty neat, eh? So the sine wave curve is what is traced out by a point that is moving circularly in the plane perpendicular to the page when you draw the 2-D representation of the spot movement on a piece of paper.

 

[berkeman]

BTW, in this 3-D visualization of the generation of a sine wave, you can label the vertical axis the Real amplitude axis, the axis straight out of the page at you the Imaginary amplitude axis, and the horizontal axis the Time axis, with positive time flowing out to the right. When you get to "complex sinusoid" representation of signals, this visualization method is a big help.