Quiz on Introduction to Inverse Trigonometric Functions

📚 Quick Study Guide

• 📐 Definition: Inverse trigonometric functions are the inverse functions of the trigonometric functions, defined over restricted domains to ensure they are one-to-one and have inverses.
• ✏️ Notation: The inverse of sine is denoted as $\arcsin(x)$ or $\sin^{-1}(x)$, cosine as $\arccos(x)$ or $\cos^{-1}(x)$, and tangent as $\arctan(x)$ or $\tan^{-1}(x)$.
• 📈 Domain and Range:
  • $\arcsin(x)$: Domain $[-1, 1]$, Range $[-\frac{\pi}{2}, \frac{\pi}{2}]$
  • $\arccos(x)$: Domain $[-1, 1]$, Range $[0, \pi]$
  • $\arctan(x)$: Domain $(-\infty, \infty)$, Range $(-\frac{\pi}{2}, \frac{\pi}{2})$
• 💡 Key Identities: Some useful identities include:
  • $\sin(\arcsin(x)) = x$ for $-1 \le x \le 1$
  • $\cos(\arccos(x)) = x$ for $-1 \le x \le 1$
  • $\tan(\arctan(x)) = x$ for all $x$
• 🧭 Understanding Principal Values: Always consider the principal value range when evaluating inverse trigonometric functions.

Practice Quiz

1. What is the value of $\arcsin(\frac{1}{2})$?
   1. $\frac{\pi}{6}$
   2. $\frac{\pi}{4}$
   3. $\frac{\pi}{3}$
   4. $\frac{\pi}{2}$
2. What is the value of $\arccos(-\frac{\sqrt{2}}{2})$?
   1. $\frac{\pi}{4}$
   2. $\frac{3\pi}{4}$
   3. $\frac{5\pi}{4}$
   4. $\frac{7\pi}{4}$
3. What is the value of $\arctan(1)$?
   1. $\frac{\pi}{6}$
   2. $\frac{\pi}{4}$
   3. $\frac{\pi}{3}$
   4. $\frac{\pi}{2}$
4. What is the domain of the function $y = \arcsin(x)$?
   1. $(-\infty, \infty)$
   2. $(-1, 1)$
   3. $[-1, 1]$
   4. $[0, \pi]$
5. What is the range of the function $y = \arccos(x)$?
   1. $(-\infty, \infty)$
   2. $(-1, 1)$
   3. $[-\frac{\pi}{2}, \frac{\pi}{2}]$
   4. $[0, \pi]$
6. Evaluate: $\sin(\arcsin(0.8))$?
   1. 0
   2. 0.6
   3. 0.8
   4. 1
7. What is the value of $\tan(\arctan(\sqrt{3}))$?
   1. 1
   2. $\sqrt{2}$
   3. $\sqrt{3}$
   4. 2