📚 Topic Summary
The inverse cosine function, denoted as $y = \arccos(x)$ or $y = cos^{-1}(x)$, provides the angle whose cosine is $x$. Its domain is $[-1, 1]$, and its range is $[0, \pi]$. Graphing the inverse cosine involves understanding its restricted domain and range, key points, and behavior. Because it's an inverse function, its graph is a reflection of the cosine function across the line $y=x$, but only for the portion of the cosine function on the interval $[0, \pi]$. Remember to accurately plot key points and understand the function's decreasing nature across its domain.
The goal of these exercises is to reinforce your understanding of how to graph the inverse cosine function. Focus on accurately plotting points and visualizing the function's key features. Understanding these concepts will make more advanced topics in trigonometry much easier.
🧮 Part A: Vocabulary
Match the terms with their definitions:
| Term |
Definition |
| 1. Inverse Cosine |
A. The set of all possible input values. |
| 2. Domain |
B. The angle whose cosine is a given number. |
| 3. Range |
C. The value the function approaches as x approaches a certain value. |
| 4. Asymptote |
D. The set of all possible output values. |
| 5. Radian |
E. A unit of angular measure equal to about 57.3 degrees. |
✍️ Part B: Fill in the Blanks
The inverse cosine function, written as $y = \arccos(x)$, has a domain of __________ and a range of __________. It returns the __________ whose cosine is $x$. The graph of $y = \arccos(x)$ is a __________ of the restricted cosine function across the line $y = x$.
🤔 Part C: Critical Thinking
Explain in your own words why the domain of the inverse cosine function is restricted to [-1, 1]. What happens if you try to input a value outside of this domain into the $\arccos(x)$ function?