Detailed Examples of Inverse Trig Functions with Restricted Domains

📚 Quick Study Guide

• 🔍 Inverse Sine (arcsin or sin^-1): The domain of $\sin^{-1}(x)$ is $[-1, 1]$, and the range is $[-\frac{\pi}{2}, \frac{\pi}{2}]$. This restriction ensures $\sin^{-1}(x)$ is a function.
• 📐 Inverse Cosine (arccos or cos^-1): The domain of $\cos^{-1}(x)$ is $[-1, 1]$, and the range is $[0, \pi]$. This restriction makes $\cos^{-1}(x)$ a function.
• 💡 Inverse Tangent (arctan or tan^-1): The domain of $\tan^{-1}(x)$ is $(-\infty, \infty)$, and the range is $(-\frac{\pi}{2}, \frac{\pi}{2})$. No restrictions on the input, but the output is limited.
• 📝 Key Formulas:
  • $\sin(\sin^{-1}(x)) = x$ for $-1 \leq x \leq 1$
  • $\cos(\cos^{-1}(x)) = x$ for $-1 \leq x \leq 1$
  • $\tan(\tan^{-1}(x)) = x$ for all $x$
  • $\sin^{-1}(\sin(x)) = x$ for $-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$
  • $\cos^{-1}(\cos(x)) = x$ for $0 \leq x \leq \pi$
  • $\tan^{-1}(\tan(x)) = x$ for $-\frac{\pi}{2} < x < \frac{\pi}{2}$
• 🧮 Understanding Restrictions: The restricted domains are crucial to ensure that the inverse trigonometric functions are indeed functions (i.e., they pass the vertical line test).

🧪 Practice Quiz

1. What is the range of the inverse sine function, $\sin^{-1}(x)$?
   1. $(0, \pi)$
   2. $[-\frac{\pi}{2}, \frac{\pi}{2}]$
   3. $(-\infty, \infty)$
   4. $[0, \frac{\pi}{2}]$
2. What is the domain of the inverse cosine function, $\cos^{-1}(x)$?
   1. $(-\infty, \infty)$
   2. $(-1, 1)$
   3. $[-1, 1]$
   4. $[0, \pi]$
3. Evaluate $\cos^{-1}(\cos(\frac{\pi}{3}))$.
   1. $\frac{\pi}{2}$
   2. $\frac{\pi}{3}$
   3. $\frac{2\pi}{3}$
   4. $\pi$
4. What is the range of the inverse tangent function, $\tan^{-1}(x)$?
   1. $[0, \pi]$
   2. $[-\frac{\pi}{2}, \frac{\pi}{2}]$
   3. $(-\frac{\pi}{2}, \frac{\pi}{2})$
   4. $(-\infty, \infty)$
5. Evaluate $\sin(\sin^{-1}(0.5))$.
   1. $\frac{\pi}{6}$
   2. $0.5$
   3. $1$
   4. $\frac{\pi}{3}$
6. What is the value of $\cos^{-1}(-1)$?
   1. $0$
   2. $\frac{\pi}{2}$
   3. $\pi$
   4. $2\pi$
7. Evaluate $\tan^{-1}(1)$.
   1. $\frac{\pi}{6}$
   2. $\frac{\pi}{4}$
   3. $\frac{\pi}{3}$
   4. $\frac{\pi}{2}$