A person taught his son about sine and cosine. He himself got introduced to them as ratios of side lengths in a right triangle, but he didn't like the idea of c

A person taught his son about sine and cosine. He himself got introduced to them as ratios of side lengths in a right triangle, but he didn't like the idea of changing definition when angles become more than 90 degrees, so he defined those functions as abscissa and ordinate of a point on a circle of unit radius, centered at origin. I think this is not perfect. Education is more of "progressing towards lesser and lesser lies", and changing definitions is an important part. The student might face it when he'll wonder about equation sin x = 2 , which will get to complex numbers. Similarly, here getting a one less power of x might correspond to "divide by x". But might sometimes not - choosing that it actually does correspond to "divide by x" is a choice. Often obvious, but sometimes not - which is seen in Gelfand's explanation of why "negative multiplied by negative makes positive", or similarly, why 0^0 is 1. Just saying that "x to one lesser power is the same divided by x" can also be seen as a convention (e.g. for some objects division can be not defined). And if it's a convention, not universal truth... then to somebody who's studying the subject this convention should be justified.