Solved Examples: Using Double-Angle Identities to Find Exact Values

📚 Quick Study Guide

• 📐 Sine Double-Angle Identity: $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$
• 📏 Cosine Double-Angle Identities: There are three common forms:
  • $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$
  • $\cos(2\theta) = 2\cos^2(\theta) - 1$
  • $\cos(2\theta) = 1 - 2\sin^2(\theta)$
• 🧭 Tangent Double-Angle Identity: $\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}$
• 💡 Tips for Solving: Choose the cosine identity that best suits the information given in the problem. Knowing the quadrant of $\theta$ helps determine the sign of trigonometric functions.
• ✍️ Remember: These identities allow you to find exact values when you know the trigonometric values of the single angle $\theta$.

Practice Quiz

1. What is the value of $\sin(2\theta)$ if $\sin(\theta) = \frac{3}{5}$ and $\cos(\theta) = \frac{4}{5}$?
   1. $\frac{12}{25}$
   2. $\frac{24}{25}$
   3. $\frac{7}{25}$
   4. $\frac{1}{5}$
2. If $\cos(\theta) = \frac{5}{13}$ and $\theta$ is in the first quadrant, what is the value of $\cos(2\theta)$?
   1. $\frac{25}{169}$
   2. $\frac{120}{169}$
   3. $\frac{119}{169}$
   4. $\frac{169}{169}$
3. Find $\tan(2\theta)$ if $\tan(\theta) = \frac{1}{2}$.
   1. $\frac{3}{4}$
   2. $\frac{4}{3}$
   3. $1$
   4. $\frac{1}{4}$
4. If $\sin(\theta) = \frac{\sqrt{3}}{2}$, what is $\cos(2\theta)$?
   1. $0$
   2. $1$
   3. $-\frac{1}{2}$
   4. $\frac{1}{2}$
5. Given that $\cos(\theta) = -\frac{1}{\sqrt{2}}$ and $\theta$ is in the second quadrant, find $\sin(2\theta)$.
   1. $1$
   2. $-1$
   3. $0$
   4. $\frac{1}{\sqrt{2}}$
6. If $\tan(\theta) = \sqrt{3}$, what is the value of $\tan(2\theta)$?
   1. $\sqrt{3}$
   2. $-\sqrt{3}$
   3. $0$
   4. Undefined
7. What is the value of $\cos(2\theta)$ if $\sin(\theta) = \frac{12}{13}$?
   1. $\frac{119}{169}$
   2. $-\frac{119}{169}$
   3. $\frac{25}{169}$
   4. $-\frac{25}{169}$