Triple Angle — Hard Practice Quiz

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

Practice quiz

1. If $\sin(3\alpha) = A$ and $\cos(3\alpha) = B$, what is the simplified expression for $A^2 + B^2$ in terms of $\alpha$?

   • $1$
   • $1 - 12\sin^2 \alpha \cos^2 \alpha$
   • $1 - 12\sin^2 \alpha \cos^2 \alpha (\sin^2 \alpha + \cos^2 \alpha)$
   • $1 - 12\sin^2 \alpha \cos^2 \alpha (\sin^2 \alpha - \cos^2 \alpha)$

   Answer: $1$

2. Express $\tan(3\alpha)$ purely in terms of $\sin \alpha$ and $\cos \alpha$, simplifying the expression obtained by substituting $\tan \alpha = \frac{\sin \alpha}{\cos \alpha}$ into the tangent triple angle formula.

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

   Answer: $\frac{3\sin \alpha \cos^2 \alpha - \sin^3 \alpha}{\cos^3 \alpha - 3\sin^2 \alpha \cos \alpha}$

3. If $\sin(3\alpha) = 0$, which of the following is a possible value for $\sin \alpha$?

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

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

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

   • $4\cos(2\alpha)$
   • $4\sin(2\alpha)$
   • $2\cos(2\alpha)$
   • $2\sin(2\alpha)$

   Answer: $4\cos(2\alpha)$

5. For which values of $\alpha$ does the denominator of the $\tan(3\alpha)$ formula, $1 - 3\tan^2 \alpha$, become zero?

   • $\alpha = n\pi + \frac{\pi}{2}$ for integer $n$
   • $\alpha = n\pi + \frac{\pi}{6}$ for integer $n$
   • $\alpha = n\pi - \frac{\pi}{6}$ for integer $n$
   • Both $\alpha = n\pi + \frac{\pi}{6}$ and $\alpha = n\pi - \frac{\pi}{6}$ for integer $n$

   Answer: Both $\alpha = n\pi + \frac{\pi}{6}$ and $\alpha = n\pi - \frac{\pi}{6}$ for integer $n$

6. If $\sin \alpha = x$, express $\cos(3\alpha)$ in terms of $x$. Assume $\alpha$ is in a quadrant where $\cos \alpha$ is positive.

   • $\sqrt{1-x^2} (1 - 4x^2)$
   • $\sqrt{1-x^2} (4x^2 - 1)$
   • $x(3 - 4x^2)$
   • $4x^3 - 3x$

   Answer: $\sqrt{1-x^2} (1 - 4x^2)$

7. If $\cos \alpha = x$, express $\sin(3\alpha)$ in terms of $x$. Assume $\alpha$ is in a quadrant where $\sin \alpha$ is positive.

   • $\sqrt{1-x^2} (4x^2 - 1)$
   • $\sqrt{1-x^2} (1 - 4x^2)$
   • $x(4x^2 - 3)$
   • $3x - 4x^3$

   Answer: $\sqrt{1-x^2} (4x^2 - 1)$

8. If $\sin(3\alpha) = k \sin \alpha$ for $\sin \alpha \neq 0$, what is $k$ in terms of $\cos \alpha$?

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

   Answer: $4\cos^2 \alpha - 1$

9. If $\cos(3\alpha) = k \cos \alpha$ for $\cos \alpha \neq 0$, what is $k$ in terms of $\sin \alpha$?

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

   Answer: $1 - 4\sin^2 \alpha$

10. Given $\sin \alpha = \frac{1}{\sqrt{10}}$ and $\cos \alpha = \frac{3}{\sqrt{10}}$, find the exact value of $\tan(3\alpha)$.

    • $\frac{13}{9}$
    • $\frac{26}{27}$
    • $\frac{1}{3}$
    • $\frac{13}{27}$

    Answer: $\frac{13}{9}$