Double Angle Identities

Practice Problems: Double Angle Identities

1. 1. What is the double angle formula for sine?
2. 2. Which of the following is NOT a valid form of $\cos(2\theta)$?
3. 3. Calculate $\sin(60°)$ using the double angle formula with $\theta = 30°$. Give your answer as a decimal rounded to 3 decimal places.
4. 4. Simplify $\cos^2(22.5°) - \sin^2(22.5°)$. Give your exact answer as a decimal rounded to 3 decimal places.
5. 5. If $\sin\theta = \frac{5}{13}$ and $\theta$ is in Quadrant I, what is $\sin(2\theta)$?
6. 6. If $\cos\theta = \frac{3}{5}$ and $\theta$ is in Quadrant IV, find $\cos(2\theta)$. Give your answer as a fraction in the form a/b.
7. 7. What is the double angle formula for tangent?
8. 8. Find $\sin(2\theta)$ if $\tan\theta = \frac{4}{3}$ and $\theta$ is in Quadrant I.
9. 9. Derive the formula $\cos(2\theta) = 1 - 2\sin^2\theta$ from $\cos(2\theta) = \cos^2\theta - \sin^2\theta$.
10. 10. Calculate $\tan(2\theta)$ if $\tan\theta = 2$. Give your answer as a fraction in the form a/b.
11. 11. Solve $\sin(2x) = \sin(x)$ for $0 \leq x < 2\pi$.
12. 12. Find $\cos(2\theta)$ if $\sin\theta = -\frac{8}{17}$ and $\theta$ is in Quadrant III.
13. 13. At what angle $\theta$ does the projectile range formula $R = \frac{v_0^2 \sin(2\theta)}{g}$ give maximum range?
14. 14. Simplify $2\cos^2(\theta) - 1$ using the double angle identity. Write your answer in terms of $\cos(2\theta)$.
15. 15. Use the power-reducing identity to express $\sin^2(x)$ in terms of $\cos(2x)$.