Trigonometric Equations — Hard Practice Quiz

A Trigonometry cheat sheet for Trigonometric Equations — every key formula with its symbols defined — plus a hard-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. Find the general solution for $x$ in the equation $2\sin^2 x - 3\sin x + 1 = 0$.

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

    Answer: $x = \frac{\pi}{6} + 2\pi n$, $x = \frac{5\pi}{6} + 2\pi n$, or $x = \frac{\pi}{2} + 2\pi n$

  2. Determine the general solution for $x$ in the equation $\cos(2x) = \sin x$.

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

    Answer: $x = \frac{\pi}{6} + 2\pi n$, $x = \frac{5\pi}{6} + 2\pi n$, or $x = \frac{3\pi}{2} + 2\pi n$

  3. Find the general solution for $x$ in $\tan x = \sin x$.

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

    Answer: $x = \pi n$

  4. If $\cos(3x) = \frac{1}{2}$ and $\sin(x) > 0$, which of the following is a possible value for $x$ in the interval $[0, 2\pi)$?

    • $\frac{11\pi}{9}$
    • $\frac{13\pi}{9}$
    • $\frac{7\pi}{9}$
    • $\frac{17\pi}{9}$

    Answer: $\frac{7\pi}{9}$

  5. What is the general solution for $x$ if $\sin(x) = \cos(x)$?

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

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

  6. Determine the general solution for $x$ in $\sin(2x) = \sin(x)$.

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

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

  7. If $\tan(x/2) = \sqrt{3}$ and $\cos x < 0$, what is the smallest positive value of $x$?

    • $\frac{\pi}{3}$
    • $\frac{2\pi}{3}$
    • $\frac{4\pi}{3}$
    • $\frac{5\pi}{3}$

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

  8. Find the general solution for $x$ in $\cos^2 x - \sin^2 x = \frac{1}{2}$.

    • $x = \pm \frac{\pi}{6} + \pi n$
    • $x = \pm \frac{\pi}{3} + \pi n$
    • $x = \pm \frac{\pi}{6} + 2\pi n$
    • $x = \pm \frac{\pi}{12} + \pi n$

    Answer: $x = \pm \frac{\pi}{6} + \pi n$

  9. Consider the equation $\sin(x) = k$. As $k$ decreases from $1$ to $0$, how does the number of distinct solutions for $x$ in the interval $[0, 2\pi)$ change?

    • It decreases from $2$ to $1$.
    • It increases from $1$ to $2$.
    • It remains constant at $2$.
    • It changes from $1$ to $0$.

    Answer: It increases from $1$ to $2$.

  10. Find the general solution for $x$ in $\sin(x) + \cos(x) = 1$.

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

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

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