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 correspond to the $y$ and $x$ coordinates respectively. - $\sec(\theta) = \frac{1}{\cos(\theta)}$ - Reference angles and signs depend on the quadrant. 3. **Step-by-step Solutions:** **(a) Calculate $\tan\left(\frac{\pi}{3}\right)$:** - $\frac{\pi}{3}$ radians is 60 degrees. - From the unit circle, $\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$ and $\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$. - Using the formula: $\tan\left(\frac{\pi}{3}\right) = \frac{\sin\left(\frac{\pi}{3}\right)}{\cos\left(\frac{\pi}{3}\right)} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3}$. **(b) Calculate $\sin\left(\frac{7\pi}{6}\right)$:** - $\frac{7\pi}{6}$ radians is 210 degrees. - This angle is in the third quadrant where sine is negative. - Reference angle is $\frac{7\pi}{6} - \pi = \frac{\pi}{6}$. - $\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$. - Therefore, $\sin\left(\frac{7\pi}{6}\right) = -\frac{1}{2}$. **(c) Calculate $\sec\left(\frac{5\pi}{3}\right)$:** - $\frac{5\pi}{3}$ radians is 300 degrees. - This angle is in the fourth quadrant where cosine is positive. - Reference angle is $2\pi - \frac{5\pi}{3} = \frac{\pi}{3}$. - $\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$. - Therefore, $\cos\left(\frac{5\pi}{3}\right) = \frac{1}{2}$. - Using the formula: $\sec\left(\frac{5\pi}{3}\right) = \frac{1}{\cos\left(\frac{5\pi}{3}\right)} = \frac{1}{\frac{1}{2}} = 2$. 4. **Final Answers:** - (a) $\tan\left(\frac{\pi}{3}\right) = \sqrt{3}$ - (b) $\sin\left(\frac{7\pi}{6}\right) = -\frac{1}{2}$ - (c) $\sec\left(\frac{5\pi}{3}\right) = 2$