## Pages

### Proof that limit of cosine theta equals 1 as theta approaches zero

This is a proof of lim_(θ -> 0) cos(θ) = 1
Consider a right angled triangle ABC with right angle at B. Let angle C be equal to θ rad.
The cosine of angle θ is defined as the ratio of the adjacent side (also called the base) and the hypotenuse:
cos(θ) = "Adjacent"/"Hypotenuse" = "BC"/"AC"
Since this proof statement has limit as angle θ approaches zero, so we will consider the above cosine ratio as angle θ approaches zero:

As angle θ gets smaller and smaller (while the other angles remain constant), the side hypotenuse and adjacent sides get more and more equal to each in length. As angle θ becomes zero, the adjacent and hypotenuse become overlapping sides and their lengths are equal.

Thus as θ -> 0, AC -> BC, so as θ = 0 AC = BC. Hence in the cosine ratio, we can write
As θ -> 0, cos(θ) = "AC"/"BC" = "BC"/"BC" or "AC"/"AC"
Thus we can write the limit of cos(θ), as θ approaches zero, as follows:
lim_(θ -> 0) cos(θ) = "BC"/"BC" = "AC"/"AC" = 1