hey all

can anyone explain why, for small $\alpha$ we may allow $\tan \alpha = \alpha$ at an intuitive, geometrical perspective. i already understand the series explanation and higher order of tangent. im just trying for a picture.

thanks!

Simon Bridge
Homework Helper
Because it is a very good approximation.

To see why: what is the slope of the tangent at ##\alpha=0##?

More exactly - look at the definition of a tangent:

The length along the tangent to a circle radius R inside the some angle ##\alpha## is ##t=R\tan\alpha##
The arclength of a circle inside the same angle ##\alpha## is ##s=R\alpha##

When R>>s, then someone standing on the surface thinks the circle is actually flat.
i.e. it looks to be the same distance as the flat tangent measure. So ##t\approx s##

1 person
PF for the win!! thanks simon

Simon Bridge