Trigonometric Equations — Practice Quiz

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

Formulas & key concepts

Solving Sine Equations

$$\sin x = a \Rightarrow x = \arcsin a + 2\pi n \text{ or } x = \pi - \arcsin a + 2\pi n$$

Where: \(n\) = integer

Solving Cosine Equations

$$\cos x = a \Rightarrow x = \pm \arccos a + 2\pi n$$

Where: \(n\) = integer

Solving Tangent Equations

$$\tan x = a \Rightarrow x = \arctan a + \pi n$$

Where: \(n\) = integer

Practice quiz

  1. What are the general solutions for $x$ in the equation $\sin x = \frac{\sqrt{3}}{2}$?

    • $x = \frac{\pi}{3} + 2\pi n \text{ or } x = \frac{2\pi}{3} + 2\pi n$
    • $x = \frac{\pi}{6} + 2\pi n \text{ or } x = \frac{5\pi}{6} + 2\pi n$
    • $x = \pm \frac{\pi}{3} + 2\pi n$
    • $x = \frac{\pi}{3} + \pi n$

    Answer: $x = \frac{\pi}{3} + 2\pi n \text{ or } x = \frac{2\pi}{3} + 2\pi n$

  2. Find the general solutions for $x$ in the equation $\cos x = -\frac{1}{2}$.

    • $x = \pm \frac{\pi}{3} + 2\pi n$
    • $x = \pm \frac{2\pi}{3} + 2\pi n$
    • $x = \frac{2\pi}{3} + 2\pi n \text{ or } x = \frac{4\pi}{3} + 2\pi n$
    • $x = \frac{5\pi}{6} + 2\pi n \text{ or } x = \frac{7\pi}{6} + 2\pi n$

    Answer: $x = \pm \frac{2\pi}{3} + 2\pi n$

  3. Determine the general solutions for $x$ in the equation $\tan x = 1$.

    • $x = \frac{\pi}{4} + 2\pi n$
    • $x = \frac{\pi}{4} + \pi n$
    • $x = \frac{3\pi}{4} + \pi n$
    • $x = \pm \frac{\pi}{4} + \pi n$

    Answer: $x = \frac{\pi}{4} + \pi n$

  4. What are all solutions for $x$ in the equation $\sin x = \frac{1}{2}$ in the interval $[0, 2\pi)$?

    • $\frac{\pi}{6}, \frac{5\pi}{6}$
    • $\frac{\pi}{3}, \frac{2\pi}{3}$
    • $\frac{\pi}{6}, \frac{7\pi}{6}$
    • $\frac{\pi}{6}$

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

  5. Find all solutions for $x$ in the equation $\cos x = 0$ in the interval $[0, 2\pi)$.

    • $\frac{\pi}{2}, \frac{3\pi}{2}$
    • $0, \pi$
    • $\frac{\pi}{2}, \frac{5\pi}{2}$
    • $\frac{\pi}{2}$

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

  6. Identify all solutions for $x$ in the equation $\tan x = -1$ in the interval $[0, 2\pi)$.

    • $\frac{3\pi}{4}, \frac{5\pi}{4}$
    • $\frac{\pi}{4}, \frac{7\pi}{4}$
    • $\frac{3\pi}{4}, \frac{7\pi}{4}$
    • $\frac{7\pi}{4}$

    Answer: $\frac{3\pi}{4}, \frac{7\pi}{4}$

  7. Solve for $x$: $2\sin x + 1 = 0$.

    • $x = \frac{\pi}{6} + 2\pi n \text{ or } x = \frac{5\pi}{6} + 2\pi n$
    • $x = -\frac{\pi}{6} + 2\pi n \text{ or } x = \frac{7\pi}{6} + 2\pi n$
    • $x = \frac{7\pi}{6} + 2\pi n \text{ or } x = \frac{11\pi}{6} + 2\pi n$
    • $x = \pm \frac{\pi}{6} + 2\pi n$

    Answer: $x = \frac{7\pi}{6} + 2\pi n \text{ or } x = \frac{11\pi}{6} + 2\pi n$

  8. What are the general solutions for $x$ in the equation $\cos(2x) = \frac{\sqrt{2}}{2}$?

    • $x = \pm \frac{\pi}{4} + 2\pi n$
    • $x = \pm \frac{\pi}{8} + 2\pi n$
    • $x = \pm \frac{\pi}{8} + \pi n$
    • $x = \frac{\pi}{8} + \pi n \text{ or } x = \frac{7\pi}{8} + \pi n$

    Answer: $x = \pm \frac{\pi}{8} + \pi n$

  9. Solve for $x$: $\tan(\frac{x}{2}) = \sqrt{3}$.

    • $x = \frac{\pi}{3} + \pi n$
    • $x = \frac{2\pi}{3} + \pi n$
    • $x = \frac{2\pi}{3} + 2\pi n$
    • $x = \frac{\pi}{6} + 2\pi n$

    Answer: $x = \frac{2\pi}{3} + 2\pi n$

  10. How many solutions does the equation $\sin x = -1$ have in the interval $[0, 4\pi)$?

    • $1$
    • $2$
    • $3$
    • $4$

    Answer: $2$

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