1. **Problem Statement:** Find the exact values of the following trigonometric functions: (a) $\tan\left(\frac{\pi}{3}\right)$ (b) $\sin\left(\frac{7\pi}{6}\right)$ (c) $\sec\left(\frac{5\pi}{3}\right)$ 2. **Formulas and Important Rules:** - $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$ - $\sin(\theta)$ and $\cos(\theta)$ values on the unit circle are based on reference angles. - $\sec(\theta) = \frac{1}{\cos(\theta)}$ - Angles are measured in radians. 3. **Step-by-step Solutions:** (a) Calculate $\tan\left(\frac{\pi}{3}\right)$: - Reference angle is $\frac{\pi}{3}$ (60 degrees). - $\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$ - $\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$ - Therefore, $\tan\left(\frac{\pi}{3}\right) = \frac{\sin\left(\frac{\pi}{3}\right)}{\cos\left(\frac{\pi}{3}\right)} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3}$ (b) Calculate $\sin\left(\frac{7\pi}{6}\right)$: - $\frac{7\pi}{6}$ is in the third quadrant where sine is negative. - Reference angle: $\frac{7\pi}{6} - \pi = \frac{\pi}{6}$ - $\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$ - So, $\sin\left(\frac{7\pi}{6}\right) = -\frac{1}{2}$ (c) Calculate $\sec\left(\frac{5\pi}{3}\right)$: - $\frac{5\pi}{3}$ is in the fourth quadrant where cosine is positive. - Reference angle: $2\pi - \frac{5\pi}{3} = \frac{\pi}{3}$ - $\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$ - $\sec\left(\frac{5\pi}{3}\right) = \frac{1}{\cos\left(\frac{5\pi}{3}\right)} = \frac{1}{\frac{1}{2}} = 2$ 4. **Final Answers:** - (a) $\tan\left(\frac{\pi}{3}\right) = \sqrt{3}$ - (b) $\sin\left(\frac{7\pi}{6}\right) = -\frac{1}{2}$ - (c) $\sec\left(\frac{5\pi}{3}\right) = 2$