Solved Examples: Applying Half-Angle Identities to Find Exact Values

📚 Quick Study Guide

🔍 Half-angle identities are used to find the trigonometric function values of angles that are half of a known angle. 💡 The half-angle identities for sine, cosine, and tangent are as follows:
• 📝 $\sin(\frac{x}{2}) = \pm \sqrt{\frac{1 - \cos(x)}{2}}$
• 🧪 $\cos(\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos(x)}{2}}$
• 📐 $\tan(\frac{x}{2}) = \pm \sqrt{\frac{1 - \cos(x)}{1 + \cos(x)}} = \frac{\sin(x)}{1 + \cos(x)} = \frac{1 - \cos(x)}{\sin(x)}$
🌍 The $\pm$ sign depends on the quadrant in which $\frac{x}{2}$ lies.

Practice Quiz

1. What is the value of $\sin(15^{\circ})$ using the half-angle identity?
   1. $\frac{\sqrt{2 - \sqrt{3}}}{2}$
   2. $\frac{\sqrt{2 + \sqrt{3}}}{2}$
   3. $\frac{\sqrt{3} - 1}{2}$
   4. $\frac{\sqrt{3} + 1}{2}$
2. What is the value of $\cos(\frac{\pi}{8})$ using the half-angle identity?
   1. $\sqrt{\frac{2 + \sqrt{2}}{2}}$
   2. $\frac{\sqrt{2 + \sqrt{2}}}{2}$
   3. $\sqrt{\frac{2 - \sqrt{2}}{2}}$
   4. $\frac{\sqrt{2 - \sqrt{2}}}{2}$
3. What is the value of $\tan(22.5^{\circ})$ using the half-angle identity?
   1. $\sqrt{2} + 1$
   2. $\sqrt{2} - 1$
   3. $1 - \sqrt{2}$
   4. $-\sqrt{2} - 1$
4. If $\cos(x) = \frac{1}{3}$ and $0 < x < \frac{\pi}{2}$, what is $\sin(\frac{x}{2})$?
   1. $\frac{\sqrt{3}}{3}$
   2. $\frac{\sqrt{6}}{3}$
   3. $\frac{\sqrt{2}}{3}$
   4. $\frac{\sqrt{5}}{3}$
5. If $\cos(x) = -\frac{1}{4}$ and $\pi < x < \frac{3\pi}{2}$, what is $\cos(\frac{x}{2})$?
   1. $\frac{\sqrt{6}}{4}$
   2. $-\frac{\sqrt{6}}{4}$
   3. $\frac{\sqrt{10}}{4}$
   4. $-\frac{\sqrt{10}}{4}$
6. Which of the following is equivalent to $\tan(\frac{x}{2})$?
   1. $\frac{1 + \cos(x)}{\sin(x)}$
   2. $\frac{\sin(x)}{1 + \cos(x)}$
   3. $\frac{1}{\cos(x)}$
   4. $\frac{1}{\sin(x)}$
7. Simplify $\sqrt{\frac{1 - \cos(2x)}{2}}$
   1. $\cos(x)$
   2. $\sin(x)$
   3. $\pm \sin(x)$
   4. $\pm \cos(x)$