Inverse Functions — Practice Quiz

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

Formulas & key concepts

Inverse Sine (Arcsine)

$$y = \arcsin x \iff x = \sin y, \quad -1 \leq x \leq 1, \quad -\frac{\pi}{2} \leq y \leq \frac{\pi}{2}$$

Inverse Cosine (Arccosine)

$$y = \arccos x \iff x = \cos y, \quad -1 \leq x \leq 1, \quad 0 \leq y \leq \pi$$

Inverse Tangent (Arctangent)

$$y = \arctan x \iff x = \tan y, \quad -\infty < x < \infty, \quad -\frac{\pi}{2} < y < \frac{\pi}{2}$$

Practice quiz

  1. What is the value of $\arcsin(\frac{\sqrt{3}}{2})$?

    • $\frac{\pi}{6}$
    • $\frac{\pi}{4}$
    • $\frac{\pi}{3}$
    • $\frac{\pi}{2}$

    Answer: $\frac{\pi}{3}$

  2. What is the value of $\arccos(-\frac{1}{2})$?

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

    Answer: $\frac{2\pi}{3}$

  3. What is the value of $\arctan(1)$?

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

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

  4. Which of the following values is NOT in the domain of $y = \arcsin x$?

    • $0.5$
    • $1$
    • $-0.7$
    • $1.2$

    Answer: $1.2$

  5. What is the range of the function $y = \arccos x$?

    • $[-\frac{\pi}{2}, \frac{\pi}{2}]$
    • $[0, \pi]$
    • $(- \frac{\pi}{2}, \frac{\pi}{2})$
    • $(0, \pi)$

    Answer: $[0, \pi]$

  6. What is the range of the function $y = \arctan x$?

    • $[-\frac{\pi}{2}, \frac{\pi}{2}]$
    • $[0, \pi]$
    • $(- \frac{\pi}{2}, \frac{\pi}{2})$
    • $(0, \pi)$

    Answer: $(- \frac{\pi}{2}, \frac{\pi}{2})$

  7. If $y = \arcsin(\frac{1}{2})$, which of the following is true?

    • $\sin y = 2$
    • $\sin y = \frac{1}{2}$
    • $\cos y = \frac{1}{2}$
    • $\tan y = \frac{1}{2}$

    Answer: $\sin y = \frac{1}{2}$

  8. Given $x = \cos(\frac{\pi}{4})$, what is the value of $\arccos x$?

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

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

  9. Evaluate $\sin(\arccos(\frac{\sqrt{2}}{2}))$?

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

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

  10. Which of the following statements is FALSE?

    • The domain of $y = \arcsin x$ is $[-1, 1]$.
    • The range of $y = \arccos x$ is $[0, \pi]$.
    • The domain of $y = \arctan x$ is $(-\infty, \infty)$.
    • The range of $y = \arcsin x$ is $(-\frac{\pi}{2}, \frac{\pi}{2})$.

    Answer: The range of $y = \arcsin x$ is $(-\frac{\pi}{2}, \frac{\pi}{2})$.

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