Double Angle — Practice Quiz

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

Practice quiz

1. If $\sin \alpha = \frac{3}{5}$ and $\cos \alpha = \frac{4}{5}$, what is the value of $\sin(2\alpha)$?

   • $\frac{12}{25}$
   • $\frac{24}{25}$
   • $\frac{7}{5}$
   • $\frac{6}{5}$

   Answer: $\frac{24}{25}$

2. Given $\cos \theta = \frac{1}{3}$, find $\cos(2\theta)$.

   • $-\frac{7}{9}$
   • $\frac{2}{3}$
   • $\frac{7}{9}$
   • $-\frac{1}{9}$

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

3. If $\tan x = 2$, what is $\tan(2x)$?

   • $-\frac{4}{3}$
   • $\frac{4}{3}$
   • $4$
   • $-\frac{2}{3}$

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

4. If $\cot \beta = 3$, what is $\cot(2\beta)$?

   • $\frac{4}{3}$
   • $\frac{3}{4}$
   • $\frac{8}{3}$
   • $\frac{3}{8}$

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

5. Simplify the expression $4\sin \theta \cos \theta$.

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

   Answer: $2\sin(2\theta)$

6. Which of the following is equivalent to $1 - 2\sin^2(3x)$?

   • $\cos(3x)$
   • $\cos(6x)$
   • $\sin(6x)$
   • $\sin(3x)$

   Answer: $\cos(6x)$

7. If $\cos(2\phi) = \frac{1}{2}$, what is the value of $2\cos^2 \phi$?

   • $\frac{1}{2}$
   • $\frac{3}{2}$
   • $\frac{1}{4}$
   • $1$

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

8. If $\tan \alpha = \frac{1}{3}$, what is the value of $\frac{2\tan \alpha}{1 - \tan^2 \alpha}$?

   • $\frac{3}{4}$
   • $\frac{1}{3}$
   • $\frac{2}{3}$
   • $\frac{3}{8}$

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

9. Which expression is equivalent to $\frac{\sin(4x)}{\cos(2x)}$?

   • $2\sin(2x)$
   • $\sin(2x)$
   • $2\cos(2x)$
   • $\tan(2x)$

   Answer: $2\sin(2x)$

10. If $\cot x = -2$, find $\cot(2x)$.

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

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