Inverse Functions — Hard Practice Quiz

A Trigonometry cheat sheet for Inverse Functions — every key formula with its symbols defined — plus a hard-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 $ \cos(\arctan(\frac{3}{4})) + \sin(\arccos(\frac{5}{13})) $?

    • $ \frac{112}{65} $
    • $ \frac{11}{13} $
    • $ \frac{17}{13} $
    • $ \frac{12}{5} $

    Answer: $ \frac{112}{65} $

  2. Solve for $ x $: $ \arctan(x) + \arctan(2x) = \frac{\pi}{4} $.

    • $ x = \frac{-3 + \sqrt{17}}{4} $
    • $ x = \frac{-3 - \sqrt{17}}{4} $
    • $ x = \frac{1}{2} $
    • $ x = \frac{1}{3} $

    Answer: $ x = \frac{-3 + \sqrt{17}}{4} $

  3. What is the domain of the function $ f(x) = \arcsin(\frac{x-1}{x+1}) $?

    • $ [0, \infty) $
    • $ (-1, \infty) $
    • $ (-\infty, -1) \cup [0, \infty) $
    • $ [-1, 1] $

    Answer: $ [0, \infty) $

  4. Which of the following statements is true regarding the principal values of inverse trigonometric functions?

    • The range of $ \arcsin x $ is $ [0, \pi] $.
    • The range of $ \arccos x $ is $ [0, \pi] $.
    • The range of $ \arctan x $ is $ [-\frac{\pi}{2}, \frac{\pi}{2}] $.
    • $ \arcsin x + \arccos x = \frac{\pi}{2} $ for all real $ x $.

    Answer: The range of $ \arccos x $ is $ [0, \pi] $

  5. Simplify the expression $ \sin(\arctan x + \arccos y) $.

    • $ \frac{xy + \sqrt{1-y^2}}{\sqrt{1+x^2}} $
    • $ \frac{x\sqrt{1-y^2} + y}{\sqrt{1+x^2}} $
    • $ \frac{xy + \sqrt{1+x^2}\sqrt{1-y^2}}{\sqrt{1+x^2}} $
    • $ \frac{x + y\sqrt{1+x^2}}{\sqrt{1+x^2}} $

    Answer: $ \frac{xy + \sqrt{1-y^2}}{\sqrt{1+x^2}} $

  6. If $ \arcsin x = \arccos(2x) $, what is the value of $ x $?

    • $ \frac{1}{\sqrt{5}} $
    • $ -\frac{1}{\sqrt{5}} $
    • $ \frac{1}{2} $
    • $ \frac{\sqrt{3}}{2} $

    Answer: $ \frac{1}{\sqrt{5}} $

  7. Consider the function $ f(x) = \arctan(x^2 - 4x + 5) $. What is the range of $ f(x) $?

    • $ [\frac{\pi}{4}, \frac{\pi}{2}) $
    • $ (-\frac{\pi}{2}, \frac{\pi}{2}) $
    • $ [0, \frac{\pi}{2}) $
    • $ [\frac{\pi}{4}, \pi) $

    Answer: $ [\frac{\pi}{4}, \frac{\pi}{2}) $

  8. If $ f(x) = \arcsin(\frac{x}{2}) $, what is $ f^{-1}(x) $?

    • $ f^{-1}(x) = 2 \sin x $, for $ x \in [-\frac{\pi}{2}, \frac{\pi}{2}] $
    • $ f^{-1}(x) = \frac{1}{2} \sin x $, for $ x \in [-1, 1] $
    • $ f^{-1}(x) = \sin(2x) $, for $ x \in [-\frac{\pi}{2}, \frac{\pi}{2}] $
    • $ f^{-1}(x) = 2 \arcsin x $, for $ x \in [-1, 1] $

    Answer: $ f^{-1}(x) = 2 \sin x $, for $ x \in [-\frac{\pi}{2}, \frac{\pi}{2}] $

  9. Evaluate $ \sin(\frac{1}{2} \arccos(\frac{1}{8})) $.

    • $ \frac{\sqrt{7}}{4} $
    • $ \frac{\sqrt{3}}{2} $
    • $ \frac{1}{4} $
    • $ \frac{\sqrt{15}}{4} $

    Answer: $ \frac{\sqrt{7}}{4} $

  10. If $ \arcsin x + \arcsin y = \frac{\pi}{2} $, and $ x, y \geq 0 $, what is the relationship between $ x $ and $ y $?

    • $ x^2 + y^2 = 1 $
    • $ x + y = 1 $
    • $ x = y $
    • $ x^2 - y^2 = 1 $

    Answer: $ x^2 + y^2 = 1 $

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