Periodicity — Hard Practice Quiz

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

Practice quiz

1. If $\sin \theta = 0.6$ and $\theta$ is an acute angle, what is the value of $\tan(\theta + 3\pi)$?

   • $\frac{3}{4}$
   • $-\frac{3}{4}$
   • $\frac{4}{3}$
   • $-\frac{4}{3}$

   Answer: $\frac{3}{4}$

2. Find all values of $x$ in the interval $[0, 4\pi]$ for which $\cos x = \cos(x + \frac{\pi}{2})$.

   • $\frac{3\pi}{4}, \frac{7\pi}{4}, \frac{11\pi}{4}, \frac{15\pi}{4}$
   • $\frac{\pi}{4}, \frac{5\pi}{4}, \frac{9\pi}{4}, \frac{13\pi}{4}$
   • $\frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \frac{7\pi}{2}$
   • No solution in the given interval

   Answer: $\frac{3\pi}{4}, \frac{7\pi}{4}, \frac{11\pi}{4}, \frac{15\pi}{4}$

3. Given $\sin \alpha = 0.8$ and $\alpha$ is in the first quadrant, what is the value of $\cos(\alpha - 5\pi)$?

   • $0.6$
   • $-0.6$
   • $0.8$
   • $-0.8$

   Answer: $-0.6$

4. Which of the following statements is true regarding the functions $f(x) = \sin(2x)$ and $g(x) = \tan(x/2)$?

   • $f(x)$ has a period of $\pi$, and $g(x)$ has a period of $2\pi$.
   • Both have a period of $\pi$.
   • $f(x)$ has a period of $2\pi$, and $g(x)$ has a period of $\pi$.
   • Both have a period of $2\pi$.

   Answer: $f(x)$ has a period of $\pi$, and $g(x)$ has a period of $2\pi$.

5. If $\alpha = \arcsin(\frac{1}{2})$, what is the value of $\cot(\alpha + \frac{7\pi}{2})$?

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

   Answer: $-\frac{1}{\sqrt{3}}$

6. If $\sin(x + 2\pi n) = \cos(y + \pi m)$ for integers $n, m$, and $x, y$ are acute angles, what is the relationship between $x$ and $y$?

   • $x = y$
   • $x + y = \frac{\pi}{2}$
   • $x - y = \frac{\pi}{2}$
   • $x + y = \pi$

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

7. Given $\cos \theta = -0.5$ and $\theta$ is in the second quadrant. What is $\sin(\theta - 6\pi) + \tan(\theta + 5\pi)$?

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

   Answer: $-\frac{\sqrt{3}}{2}$

8. If the period of $f(x) = \sin(ax)$ is $P_1$ and the period of $g(x) = \tan(bx)$ is $P_2$. If $P_1 = P_2$ and $a, b > 0$, what is the ratio $a/b$?

   • $1/2$
   • $1$
   • $2$
   • $4$

   Answer: $2$

9. Evaluate $\frac{\sin(\frac{13\pi}{6})}{\cos(\frac{17\pi}{3})}$.

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

   Answer: $1$

10. Consider a function $f(x)$ such that $f(x) = f(x + 2\pi)$ for all $x$. Which of the following statements must be true?

    • $f(x)$ must have a fundamental period of $2\pi$.
    • $f(x)$ cannot be $\tan x$.
    • $f(x)$ could have a fundamental period of $\pi$.
    • $f(x)$ must be $\sin x$ or $\cos x$.

    Answer: $f(x)$ could have a fundamental period of $\pi$.