Triple Angle — Practice Quiz

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

Practice quiz

1. If $\sin \alpha = \frac{1}{3}$, what is the value of $\sin(3\alpha)$?

   • $\frac{23}{27}$
   • $\frac{1}{27}$
   • $\frac{25}{27}$
   • $\frac{2}{3}$

   Answer: $\frac{23}{27}$

2. Given $\cos \alpha = \frac{1}{2}$, find the value of $\cos(3\alpha)$?

   • $-1$
   • $\frac{1}{2}$
   • $\frac{\sqrt{3}}{2}$
   • $0$

   Answer: $-1$

3. If $\tan \alpha = 1$, what is $\tan(3\alpha)$?

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

   Answer: $-1$

4. Simplify the expression $\frac{\sin(3\alpha)}{\sin \alpha}$ for $\sin \alpha \neq 0$.

   • $3 - 4\sin^2 \alpha$
   • $3 + 4\sin^2 \alpha$
   • $4\sin^2 \alpha - 3$
   • $1 - 4\cos^2 \alpha$

   Answer: $3 - 4\sin^2 \alpha$

5. Which of the following is equivalent to $\cos(3\alpha)$?

   • $4\cos^3 \alpha - 3\cos \alpha$
   • $3\cos \alpha - 4\cos^3 \alpha$
   • $3\sin \alpha - 4\sin^3 \alpha$
   • $1 - 2\sin^2(3\alpha)$

   Answer: $4\cos^3 \alpha - 3\cos \alpha$

6. If $\tan(3\alpha) = \frac{3\tan \alpha - \tan^3 \alpha}{X}$, what is $X$?

   • $1 - 3\tan^2 \alpha$
   • $1 + 3\tan^2 \alpha$
   • $3\tan^2 \alpha - 1$
   • $1 - \tan^2 \alpha$

   Answer: $1 - 3\tan^2 \alpha$

7. Evaluate $\sin(90^\circ)$ using the triple angle formula for sine, given $\sin(30^\circ) = \frac{1}{2}$.

   • $1$
   • $\frac{\sqrt{3}}{2}$
   • $\frac{1}{2}$
   • $0$

   Answer: $1$

8. If $\cos \alpha = \frac{\sqrt{3}}{2}$, what is $\sin(3\alpha)$?

   • $1$
   • $\frac{\sqrt{3}}{2}$
   • $\frac{1}{2}$
   • $0$

   Answer: $1$

9. Which of the following expressions is equivalent to $\frac{\cos(3\alpha)}{\cos \alpha}$ for $\cos \alpha \neq 0$?

   • $4\cos^2 \alpha - 3$
   • $3 - 4\cos^2 \alpha$
   • $4\sin^2 \alpha + 1$
   • $1 + 4\sin^2 \alpha$

   Answer: $4\cos^2 \alpha - 3$

10. If $\tan \alpha = \frac{1}{\sqrt{3}}$, what is $\tan(3\alpha)$?

    • Undefined
    • $0$
    • $\sqrt{3}$
    • $1$

    Answer: Undefined