Proving Small Angle Theorems

Consider the AREAS in the following diagram

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)

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

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

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