As θ takes every real value, the expression 3 + 2sin(2θ − 40°) takes exactly the values:

As θ runs over every real value, the inner angle 2θ − 40° also runs over every angle, so sin(2θ − 40°) sweeps the whole interval −1 to 1. Multiplying by 2 gives −2 to 2 and adding 3 shifts that to 1 to 5, with both ends attained. The doubling and the −40° shift change only how quickly and where the sweep starts, never how far it reaches. Ignoring the +3 leaves −2 to 2, and using only the non-negative part of the sine gives 3 to 5.