7.1.1: Resources and Key Concepts

Prerequisite Topics

This page lists the prerequisite skills (all of which can be reviewed in CRC's Corequisite Codex) needed for this section that have not already been mentioned in any previous section. If you are enrolled in a course with a Support section, some (but definitely not all) of these topics might be covered or reviewed in the Support section of your course.

• An Overview of Functions
  • Function Composition and Domain: This skill is necessary for evaluating and simplifying composite functions involving trigonometric and inverse trigonometric functions, such as \( \sin(\arccos(x)) \).
  • Inverse Functions: This section is built on the concept of inverse functions. Finding a formula for an inverse, using the Horizontal Line Test to determine if a function has an inverse function, and understanding the relationship between the domain and range of a function and its inverse are all critical prerequisite skills. These are reviewed in the "Skills Refresher" exercises.
  • Restricting Domains and Forcing Invertibility: This section's primary theoretical basis is the idea of restricting the domains of the non-invertible trigonometric functions to create new functions that are one-to-one and thus have inverse functions.

Videos

• Inverse Trigonometric Functions
  • Implicit Differentiation - Derivatives of Inverse Trig Functions: Theorem, Proof, and Example
  • Inverse Trigonometric Functions - Defining the Inverse Trigonometric Functions
  • Inverse Trigonometric Functions - Evaluating Inverse Trigonometric Functions
  • Inverse Trigonometric Functions - Applications of Right Triangle Trigonometry (Part 2)

Definitions

• The Inverse Sine Function: The function \(f(x) = \sin^{-1}(x)\) (or \(\arcsin(x)\)) is defined as \(\sin^{-1}(x) = \theta\) if and only if \(\sin(\theta) = x\) and \(-\frac{\pi}{2} \le \theta \le \frac{\pi}{2}\).
• The Inverse Cosine Function: The function \(f(x) = \cos^{-1}(x)\) (or \(\arccos(x)\)) is defined as \(\cos^{-1}(x) = \theta\) if and only if \(\cos(\theta) = x\) and \(0 \le \theta \le \pi\).
• The Inverse Tangent Function: The function \(f(x) = \tan^{-1}(x)\) (or \(\arctan(x)\)) is defined as \(\tan^{-1}(x) = \theta\) if and only if \(\tan(\theta) = x\) and \(-\frac{\pi}{2} < \theta < \frac{\pi}{2}\).

Theorems

• Lemma: Let \(\theta\) be an angle in standard position, and suppose the terminal side of \(\theta\) intersects the unit circle at the point \(P(x, y)\). Then \(\cos(\theta) = x\) and \(\sin(\theta) = y\).
• Forcing Invertibility by Restricting Domain: If we restrict the domain of a non-invertible function so that the resulting function becomes one-to-one, then this new function will have an inverse.
• Properties of the Inverse Sine Function: For \(f(x) = \sin^{-1}(x)\):
  • Domain: \([-1, 1]\)
  • Range: \([-\frac{\pi}{2}, \frac{\pi}{2}]\)
  • \(\sin(\sin^{-1}(x)) = x\) for all \(x\) in \([-1, 1]\)
  • \(\sin^{-1}(\sin(x)) = x\) provided \(x\) is in \([-\frac{\pi}{2}, \frac{\pi}{2}]\)
• Properties of the Inverse Cosine Function: For \(f(x) = \cos^{-1}(x)\):
  • Domain: \([-1, 1]\)
  • Range: \([0, \pi]\)
  • \(\cos(\cos^{-1}(x)) = x\) for all \(x\) in \([-1, 1]\)
  • \(\cos^{-1}(\cos(x)) = x\) provided \(x\) is in \([0, \pi]\)
• Properties of the Inverse Tangent Function: For \(f(x) = \tan^{-1}(x)\):
  • Domain: \((-\infty, \infty)\)
  • Range: \((-\frac{\pi}{2}, \frac{\pi}{2})\)
  • \(\tan(\tan^{-1}(x)) = x\) for all \(x\) in \(\mathbb{R}\)
  • \(\tan^{-1}(\tan(x)) = x\) provided \(x\) is in \((-\frac{\pi}{2}, \frac{\pi}{2})\)

Common Mistakes

• Inverse Composition Property: Be cautious with compositions like \(\sin^{-1}(\sin(\theta))\). This expression is not necessarily equal to \(\theta\). It is only equal to \(\theta\) if \(\theta\) is within the restricted range of the inverse sine function, \([-\frac{\pi}{2}, \frac{\pi}{2}]\). A similar caution applies to inverse cosine and inverse tangent.
• Inverse Trigonometric Functions are NOT Reciprocals: The notation \(\sin^{-1}(x)\) denotes the inverse sine function, not the reciprocal of sine. The reciprocal of \(\sin(x)\) is \(\csc(x)\).
• Forgetting Domain and Range Restrictions: The inverse trigonometric functions have specific, restricted ranges. For example, the output of \(\arccos(x)\) must be an angle between \(0\) and \(\pi\), inclusive. Forgetting these restrictions is a common source of error.