1. Proving Small Angle Theorems

2. Consider the AREAS in the following diagram

3. Thinking about AREAS
Triangle OAB ≤ Sector OAB ≤ Triangle OAC ½ r2Sin(θ) ≤ ½r2θ ≤ ½r2 tan(θ) 1 ≤ θ sin (θ) ≤ tan(θ) sin(θ) 1 ≤ θ sin (θ) ≤ cos(θ)
Now as θ → 0, cos(θ )→ 1
≤ θ sin (θ) ≤ 1 𝑛𝑒𝑎𝑟𝑙𝑦!
So as θ → 0, θ = sin (θ)

4. Tan(θ)
1 ≤ θ sin (θ) ≤ cos(θ) 1 cos⁡(θ) ≤ θ tan (θ) ≤ 1 Then as before…

5. Cos(θ)
As θ→0, cos (θ) → 1
There is a better approximation which will be proved later (double angle formulea)
𝐶𝑜𝑠 𝜃 =1− 𝜃 2 2

6. Summary For small angles (-0.1<θ<0.1 radians) Sin(θ) = θ
Tan(θ) = θ
Cosθ = 1− 𝜃 2 2