As θ varies over all real values, the greatest value of 1/(3 − 2cosθ) is:

Since cosθ runs from −1 to 1, the denominator 3 − 2cosθ runs from 1 to 5 and is never zero. A fraction with a fixed numerator is largest when its denominator is smallest, so the maximum is 1/1 = 1 at cosθ = 1. Choosing cosθ = −1 makes the denominator biggest and gives the minimum 1/5, and cosθ = 0 gives the mid value 1/3.