Restricting the Domain
Trig functions are periodic — they repeat forever. To “undo” them with an inverse, we must first restrict their domains so each output maps to exactly one input.
Introduction
In algebra, you learned that a function must pass the horizontal line test to have an inverse. Sine, cosine, and tangent all fail this test — every output value is hit infinitely many times. The solution: restrict the domain to a carefully chosen interval where the function is one-to-one.
Past Knowledge
Inverse functions, horizontal line test, and graphs of sin, cos, tan from Chapter 4.
Today's Goal
Identify the restricted domains for arcsin, arccos, and arctan and evaluate inverse trig expressions.
Future Success
Every equation in Lessons 6.2–6.8 requires understanding restricted domains to find correct solutions.
Key Concepts
Why Restrict?
, but so does , , etc. If we ask “what angle has sine equal to ?” there are infinitely many answers. To get a unique answer, we restrict each function to an interval where it hits every output value exactly once.
The Restricted Domains
| Inverse Function | Restricted Domain of Original | Range of Inverse |
|---|---|---|
Key Insight
The restricted domain of the original function becomes the range of the inverse. When you evaluate , your answer must fall in — no exceptions.
Visualizing the Restriction
Below, the full sine curve is shown in light gray. The restricted portion (the part that becomes the inverse) is highlighted in blue:
Worked Examples
Evaluating Inverse Sine
Question: Evaluate .
Step 1: Ask: “What angle in has sine equal to ?”
Step 2: , and ✓
Final Answer:
Evaluating Inverse Cosine
Question: Evaluate .
Step 1: Ask: “What angle in has cosine equal to ?”
Step 2: , and ✓
Final Answer:
Composition Trap
Question: Evaluate .
Step 1: Compute the inner function:
Step 2: Now evaluate:
Key: The answer is , not , because is outside the range of arcsin.
Final Answer:
Common Pitfalls
Giving Answers Outside the Range
even though . The answer must be in the restricted range. For arcsin, that's .
Confusing the Three Ranges
Arcsin returns angles in QI/QIV, arccos in QI/QII, and arctan in QI/QIV. Mixing these up is the most common source of error in this chapter.
Real-Life Applications
GPS & Navigation Systems
When a GPS receiver calculates your position, it solves trig equations involving satellite angles. The restricted domain ensures the system returns a single, unambiguous latitude and longitude — without it, the algorithm wouldn't know which of the infinitely many solutions is your actual location.
Practice Quiz
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