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Can anyone, please, help me with the concept of defining trigonometric functions? I don't understand what is written here:

enter image description here

As far as I remember from school, the sine is the ratio of the opposite leg (of an angle in the right triangle) to hypotenuse. However, in the image above I don't see any right triangles. I don't even see any triangles at all! How is it then that "we define that sine of t is equal to y"?

I seem to be missing some important link here and, therefore, cannot grasp the basics of the concept of definition of trigonometric functions.

brilliant
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  • Useful google image search -- trigonometry + unit + circle + triangle – Dave L. Renfro Oct 02 '22 at 09:40
  • @DaveL.Renfro - Bunch of images, but no explanations. – brilliant Oct 02 '22 at 09:45
  • The answer is that we generalize: we start in a space where one principle holds but then consider a more general form of that principle that holds in a larger space encompassing the first. This approach is ubiquitous in math. We start from the set of right angled triangles with unit hypotenuse, defining $\sin()$, $\cos()$ as its side-lengths. But if we recognize that the hypotenuse can be treated as a unit radius in Quadrant I, we can keep it turning into the other quadrants, now treating $\sin()$, $\cos()$ as the location of its endpoint. This naturally generalizes the first definition. – Jam Oct 02 '22 at 10:04
  • @Joe - Only for cases in quadrant 1. – brilliant Oct 02 '22 at 11:02
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    Bunch of images, but no explanations. --- Plenty of explanations if you click on the images and visit their various originating pages. Regarding the answer by @Joe in the linked question (nice, so +1), of course it's only for cases in quadrant 1 since the right triangle trig relations only correspond to quadrant 1 angles. Did you miss his analogy with defining exponents that are not positive integers? – Dave L. Renfro Oct 02 '22 at 13:34

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