Reversing the Process

Trigonometry · Axiom Academy

If we know the sine value, can we find the angle? Let's discover why this question is trickier than it seems! First, let's remember how sine works: we put in an angle and get a sine value . Move the slider to explore! Enter any sine value between -1 and 1, and let's find all the angles in one revolution (0° to 360°) that produce it. ✂ The Solution: Pick One Slice To create an inverse function, we need to restrict sine to a domain where each sine value corresponds to exactly one angle. The Chosen Domain: -90° to 90° This range captures all sine values exactly once! Test it! Enter a sine value and see which angle is chosen: Understanding Inverse Functions Sine is a periodic function that repeats every 360°. Without restriction, infinitely many angles produce the same sine value. This makes it impossible to define a proper inverse function. We restrict sine's domain to [-90°, 90°] (or [-π/2, π/2] in radians). In this range, sine is one-to-one: each output corresponds to exactly one input. This restricted version can be inverted! The inverse sine function (arcsin or sin⁻¹) always returns an angle in the range [-90°, 90°]. This is called the principal value . While other angles might have the same sine, arcsin gives us the unique answer in this standard range. The same issue affects cosine and tangent! Each requires its own domain restriction to create an inverse function. Understanding why we need these restrictions helps you work confidently with inverse trig functions.

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