Understanding Inverse Trigonometric Functions and Their Domains

Miacademy & MiaPrep Learning Channel

This comprehensive mathematics video explores the concept of inverse trigonometric functions, specifically focusing on inverse sine, cosine, and tangent. The lesson begins by addressing the fundamental problem that trigonometric functions are periodic and fail the horizontal line test, meaning they are not naturally one-to-one. The narrator explains how to restrict the domains of these functions to specific intervals—such as [-pi/2, pi/2] for sine—to create invertible segments, allowing for the definition of arcsin, arccos, and arctan. The video provides a deep dive into visualizing these concepts using graphs and the unit circle. It demonstrates how inverse functions are reflections over the line y=x and clearly defines the new domain and range for each inverse function. Through multiple worked examples, students learn how to evaluate inverse trigonometric expressions both by finding exact values on the unit circle (for special angles) and by using a calculator for approximations. The content also addresses common pitfalls, such as why evaluating the inverse cosine of 2 results in an undefined answer. For educators, this video is an excellent resource for Precalculus or Trigonometry units. It visualizes abstract concepts like domain restriction and function reflection, making them accessible to students. The clear step-by-step examples provide models for solving problems without a calculator, reinforcing unit circle fluency, while also teaching proper calculator usage for non-standard angles. The video serves as a crucial bridge between basic trigonometry and solving trigonometric equations.