Why Restrict the Domain?

Trigonometry · Axiom Academy

LESSON Why Restrict the Domain? Understanding why trigonometric functions need restricted domains for their inverses to exist 1. The Problem with Periodic Functions Consider the sine function. If we ask "what angle has a sine of 0.5?", there are infinitely many correct answers: 30°, 150°, 390°, 510°, and so on. This violates a fundamental requirement for inverse functions. For a function to have an inverse, it must be one-to-one : each output must correspond to exactly one input. We can check this with the horizontal line test —if any horizontal line intersects the graph more than once, the function is not one-to-one. The solution is to restrict the domain—choose a continuous interval where the function is one-to-one. For sine, we choose [-π/2, π/2] (or [-90°, 90°]). On this interval, sine increases from -1 to 1 without repeating any values. 4. Standard Restricted Domains Each trigonometric function has a standard restricted domain chosen to: Include the complete range of output values Be as simple and symmetric as possible

This is the written version of the interactive lesson above. See the full Trigonometry course.