Inverse Function Relations — Practice Quiz

A Trigonometry cheat sheet for Inverse Function Relations — every key formula with its symbols defined — plus a medium-level practice quiz to test recall.

Practice quiz

1. Evaluate $\arcsin(-\frac{1}{2})$.

   • $-\frac{\pi}{6}$
   • $\frac{\pi}{6}$
   • $\frac{5\pi}{6}$
   • $-\frac{\pi}{3}$

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

2. Evaluate $\arccos(-\frac{\sqrt{3}}{2})$.

   • $\frac{\pi}{6}$
   • $-\frac{\pi}{6}$
   • $\frac{5\pi}{6}$
   • $\frac{2\pi}{3}$

   Answer: $\frac{5\pi}{6}$

3. What is the value of $\arcsin(\frac{1}{5}) + \arccos(\frac{1}{5})$?

   • $\frac{\pi}{2}$
   • $\pi$
   • $0$
   • $1$

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

4. If $\arctan x = \frac{\pi}{9}$, what is $\text{arccot } x$?

   • $\frac{7\pi}{18}$
   • $\frac{\pi}{9}$
   • $\frac{8\pi}{9}$
   • $\frac{\pi}{2}$

   Answer: $\frac{7\pi}{18}$

5. Simplify the expression $\arcsin(-x) + \arccos(x)$.

   • $\frac{\pi}{2} - 2\arcsin x$
   • $\frac{\pi}{2}$
   • $-\frac{\pi}{2}$
   • $0$

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

6. Simplify $\arcsin(-x) + \arccos(-x)$.

   • $\frac{\pi}{2}$
   • $\pi - 2\arcsin x$
   • $\pi$
   • $0$

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

7. If $\arcsin x = \frac{\pi}{4}$, what is $\arccos x$?

   • $\frac{\pi}{4}$
   • $\frac{\pi}{2}$
   • $0$
   • $-\frac{\pi}{4}$

   Answer: $\frac{\pi}{4}$

8. Given $\text{arccot } x = \frac{3\pi}{7}$, find $\arctan x$.

   • $\frac{\pi}{14}$
   • $\frac{3\pi}{7}$
   • $\frac{4\pi}{7}$
   • $\frac{\pi}{2}$

   Answer: $\frac{\pi}{14}$

9. Which of the following statements is true for all valid $x$?

   • $\arcsin(-x) = \arcsin x$
   • $\arccos(-x) = -\arccos x$
   • $\arcsin(-x) = -\arcsin x$
   • $\arctan x + \text{arccot } x = \pi$

   Answer: $\arcsin(-x) = -\arcsin x$

10. Evaluate $\arcsin(-\frac{\sqrt{2}}{2}) + \arccos(-\frac{\sqrt{2}}{2})$.

    • $\frac{\pi}{2}$
    • $\frac{3\pi}{4}$
    • $\frac{\pi}{4}$
    • $\pi$

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