Inverses of trigonometric functions

Learning Objectives Understand why trigonometric functions require domain restrictions to have inverses. Define the domain and range of arcsin, arccos, and arctan. Evaluate inverse trigonometric expressions and find principal values. Simplify composite expressions involving trigonometric and inverse trigonometric functions. By the end of of this lesson, students will be able to solve trigonometric equations using inverse functions. Calculate the derivatives of functions involving inverse trigonometric functions. If you know a ramp is 10 meters long and rises 2 meters high, how can you find the exact angle of inclination? 🤔 Inverse trig functions are your answer! This tutorial explores the inverse trigonometric functions: arcsine, arccosine, and arctangent. We will learn ho...

TermDefinitionExample One-to-One FunctionA function where every output value is associated with exactly one input value. A function must be one-to-one to have an inverse. Graphically, it must pass the Horizontal Line Test.f(x) = x³ is one-to-one. f(x) = x² is not, because f(2) = 4 and f(-2) = 4. Domain RestrictionThe process of limiting the domain of a function that is not one-to-one, in order to create a new function that is one-to-one and thus has a well-defined inverse.The function y = sin(x) is not one-to-one. By restricting its domain to [-π/2, π/2], the new function has an inverse, y = sin⁻¹(x). Principal ValueThe unique output value of an inverse trigonometric function that falls within its restricted range.While there are infinite angles whose sine is 1/2 (e.g., π/6, 5π/6, 13π/6),...

Inverse Composition Properties 1. sin(sin⁻¹(x)) = x for x in [-1, 1] 2. cos(cos⁻¹(x)) = x for x in [-1, 1] 3. tan(tan⁻¹(x)) = x for all real x 4. sin⁻¹(sin(x)) = x for x in [-π/2, π/2] 5. cos⁻¹(cos(x)) = x for x in [0, π] 6. tan⁻¹(tan(x)) = x for x in (-π/2, π/2) These rules define how trigonometric functions and their inverses 'cancel' each other out. The validity of the cancellation depends on whether the variable 'x' is in the domain of the inner function and whether the result is in the range of the outer function. Be very careful with the domain restrictions for rules 4, 5, and 6. Derivatives of Inverse Trigonometric Functions 1. d/dx (sin⁻¹(u)) = (1 / √(1 - u²)) * du/dx 2. d/dx (cos⁻¹(u)) = (-1 / √(1 - u²)) * du/dx 3. d/dx (tan⁻¹(u)) = (1 / (1 + u²))...