Cancellation Properties

Trigonometry · Axiom Academy

LESSON Cancellation Properties Understanding when sin(arcsin(x)) = x and when arcsin(sin(x)) = x through domain and range considerations When we compose a function with its inverse, we expect to get back our original input. For sine and arcsine, this means: The answer depends on the domain and range of these functions. Let's visualize how these compositions work. This composition works perfectly when x is in the domain of arcsin, which is the set of values we can plug into arcsin without getting an error. Why? The domain of arcsin is [−1, 1]. If x is outside this interval, arcsin(x) is undefined, so the composition doesn't exist. But for any x in [−1, 1], arcsin returns an angle in [−π/2, π/2], and sin of that angle gives us back x. This composition is trickier! It only returns x when x is in the range of arcsin, not just any angle. Why the restriction? The range of arcsin is [−π/2, π/2]. When you compute sin(x), you get a value in [−1, 1], which is fine. But then arcsin returns the principal angle with that sine value—always in [−π/2, π/2]. This is only equal to your original x if x was already in that range! Let's see both compositions side by side to understand why they have different requirements: What happens when x is outside [−π/2, π/2]? We can still compute arcsin(sin(x)), but it won't equal x. Instead, arcsin returns an equivalent angle in the principal range. If x ∈ [−π/2, π/2]: arcsin(sin(x)) = x If x ∈ (π/2, π]: arcsin(sin(x)) = π − x

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