- #1

PFuser1232

- 479

- 20

Almost every "geometric proof" I see for formulas like the one I mentioned above, or results such as ##\lim_{x \rightarrow 0} \frac{\sin{x}}{x} = 1##, places a restriction (usually ##[0,\frac{\pi}{2}]## on the argument of the sine (or cosine, or tangent) function. The result is then said to be true for all real numbers.

I've seen this over and over again in physics as well. When there is, for example, some force that varies with time, we express the components of that force in terms of the angles given by looking at a convenient case where some right triangle is formed, before applying rules from "classic" geometry such as "the sum of the angles in a triangle is always ##\pi##". We then assume the results we get hold true for

**all**angles.

Even when deriving the equations of motion in polar coordinates, while the "analytical derivation" seems to work for angles and scalar components of all magnitudes and signs, the "geometric derivation" almost always involves looking at some convenient situation where the angle/increment in angle is always positive, and where the scalar components of a vector (which can be positive, negative, or zero) are treated as magnitudes.