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At first, trig ratios related only to right triangles. Then you learned how to find ratios for any angle, using all four quadrants. Then you learned about the unit circle, in which the value of the hypotenuse was always r = 1 so that sin(θ) = y and cos(θ) = x. In other words, you progressed from geometrical figures to a situation in which there was just one input (one angle measure, instead of three sides and an angle) leading to one output (the value of the trig ratio). And this kind of relationship can be turned into a function.

Looking at the sine ratio in the four quadrants, we can take the input (the angle measure θ), "unwind" this from the unit circle, and put it on the horizontal axis of a standard graph in the x,y-plane. Then we can take the output (the value of sin(θ) = y) and use this value as the height of the function. The result looks like this:

Sineanim

Sine Animations






As you can see, the height of the red line, being the value of sin(θ) = y, is the same in each graph. In the unit circle on the left, the angle is indicated by the green line. On the "regular" graph on the right, the angle is indicated by the scale on the horizontal axis.

If the green angle line had gone backwards, counting into negative angle measures, the horizontal graph on the right would have extended back to the left of zero. If, instead of starting over again at zero for every revolution on the unit circle, we'd counted up higher angles, then the horizontal graph on the right would have continued, up and down, over and over again, past 2π.

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