Famous Trig Limit

The important trigonometric limit: sin(θ)/θ -> 1 as θ -> 0 depends on our using radian measure for θ, as can be seen from the derivation, below, where the area of the wedge, 0.5*θ, is only valid for θ measured in radians.

Slide the point on the unit circle, and observe that the wedge is trapped between the two triangles. A bit of algebra, and the squeeze theorem then yields the result.

It is good to remember the basic set-up of this proof: The point on the circle is: (cos(θ), sin(θ)), and the point of intersection with x = 1 is the point (1, tan(θ)). Notice also that the secant is present in the length of the hypotenuse of the larger triangle!

Dave Matthews, Created with GeoGebra