Inverse Function Relations — Hard Practice Quiz

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

Formulas & key concepts

Arcsine Odd Function

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

Arccosine Symmetry

$$\arccos(-x) = \pi - \arccos x$$

Complementary Inverses

$$\arcsin x + \arccos x = \frac{\pi}{2}$$

Arctangent and Arccotangent

$$\arctan x + \text{arccot } x = \frac{\pi}{2}$$

Practice quiz

  1. Evaluate the expression $\arcsin(-x) + \arccos(-x)$.

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

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

  2. If $\arcsin x = A$, express $\arccos(-x)$ in terms of $A$.

    • $\frac{\pi}{2} + A$
    • $\frac{\pi}{2} - A$
    • $\pi - A$
    • $\pi + A$

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

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

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

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

  4. If $\arcsin(-x) = \frac{\pi}{6}$, what is the value of $\arccos x$?

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

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

  5. Given $\arccos(-x) = \frac{2\pi}{3}$, find $\arcsin x$.

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

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

  6. If $\arctan x = \frac{\pi}{3}$ and $\text{arccot } y = \frac{\pi}{6}$, what is the value of $\arctan y + \text{arccot } x$?

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

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

  7. If $\arcsin x = \frac{\pi}{4}$, what is the value of $\arccos(-x) - \arcsin(-x)$?

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

    Answer: $\pi$

  8. Given $\arcsin x + \arccos(-x) = \frac{3\pi}{4}$, find $\arcsin x$.

    • $\frac{\pi}{8}$
    • $\frac{\pi}{4}$
    • $\frac{3\pi}{8}$
    • $\frac{\pi}{2}$

    Answer: $\frac{\pi}{8}$

  9. If $\arccos x = \frac{\pi}{5}$, what is the value of $\arcsin(-x)$?

    • $-\frac{3\pi}{10}$
    • $\frac{3\pi}{10}$
    • $-\frac{\pi}{5}$
    • $\frac{\pi}{5}$

    Answer: $-\frac{3\pi}{10}$

  10. Consider the equation $\arcsin x + \arccos x = k$. If $x$ is replaced by $-x$, how does the value of $k$ change?

    • It remains unchanged.
    • It becomes $\pi - k$.
    • It becomes $k - \pi$.
    • It becomes $2k$.

    Answer: It remains unchanged.

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