Limit of sin(x) = x (as x tends towards 0)
Consider this diagram. There are three shapes drawn about the unit circle and the angle POA = x
1. Area of Triangle OBA = base x height - now base = 1 and tan(x) = since OA = 1 height = tan(x)
Area = tan(x)
2. Area of Triangle OPA = ab sinC - now a and b = 1 (since it is the unit circle)
Area = sin(x)
3. Area of the sector = the fraction of the circle x the area of the circle.. And since r =1
Area =
The area of the sector is obviously between the areas of the triangles. Therefore
We can multiply everything by 2
Now tan (x) is sin (x) divided by cos (x)
Now as x tends towards zero. cos(x) tends towards 1. So we get
Therefore using this squeeze principle as x tends toward zero sin(x) = x