Why does the tangent function behave like the angle itself on a unit circle?

[member 428835]

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. I am just trying for a picture.

thanks!

 

[Simon Bridge]

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 $t\approx s$

 

[member 428835]

PF for the win! thanks simon

 

[Simon Bridge]

Cool!

By the same token:
sinA = A
cosA = 1

When you realize that the trig functions are the names of lengths defined on a unit circle the whole thing makes a lot more sense.