Let's solve each expression in turn.

(a) $\sin^{-1}(-1)$

The inverse sine function, $\sin^{-1}(x)$, has a range of $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ and is defined for values of $x$ between -1 and 1. For $\sin^{-1}(-1)$, we are looking for the angle $\theta$ such that: $\sin(\theta) = -1$ In the interval $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$, the angle that satisfies this condition is $\theta = -\frac{\pi}{2}$.

Answer: $-\frac{\pi}{2}$

(b) $\cos^{-1}(0)$

The inverse cosine function, $\cos^{-1}(x)$, has a range of $[0, \pi]$ and is defined for values of $x$ between -1 and 1. For $\cos^{-1}(0)$, we are looking for the angle $\theta$ such that: $\cos(\theta) = 0$ In the interval $[0, \pi]$, the angle that satisfies this condition is $\theta = \frac{\pi}{2}$.

Answer: $\frac{\pi}{2}$

(c) $\tan^{-1}\left(\frac{\sqrt{3}}{3}\right)$

The inverse tangent function, $\tan^{-1}(x)$, has a range of $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ and is defined for all real values of $x$. For $\tan^{-1}\left(\frac{\sqrt{3}}{3}\right)$, we are looking for the angle $\theta$ such that: $\tan(\theta) = \frac{\sqrt{3}}{3}$ The angle that satisfies this condition in the range $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ is $\theta = \frac{\pi}{6}$.

Answer: $\frac{\pi}{6}$

Summary of Answers:

• (a) $-\frac{\pi}{2}$
• (b) $\frac{\pi}{2}$
• (c) $\frac{\pi}{6}$

Would you like a detailed explanation for any of these steps?

Here are some related questions you might find useful to explore:

1. What are the ranges of inverse trigonometric functions and why are they restricted?
2. How do we determine if an inverse trigonometric expression is undefined?
3. How can we use reference angles to find the exact values of trigonometric functions?
4. What are the practical applications of inverse trigonometric functions in real-world scenarios?
5. How can we convert radians to degrees in trigonometric calculations?

Tip: Remember that the outputs of inverse trigonometric functions fall within specific intervals. This helps to identify correct angles.