2) Using a diagram determine the 6 trigonometric ratios for the angle (7pi )/(6) (do not rationalize the denominator)

## Step 1: Determine the Reference Angle<br />### The angle \( \frac{7\pi}{6} \) is in the third quadrant. To find the reference angle, subtract \( \pi \) from \( \frac{7\pi}{6} \). This gives us a reference angle of \( \frac{\pi}{6} \).<br />## Step 2: Identify the Signs of Trigonometric Functions<br />### In the third quadrant, sine and cosine are negative, while tangent is positive. Therefore, \(\sin\), \(\cos\), \(\csc\), and \(\sec\) will be negative, and \(\tan\) and \(\cot\) will be positive.<br />## Step 3: Calculate Sine and Cosecant<br />### For the reference angle \( \frac{\pi}{6} \), \(\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\). Thus, \(\sin\left(\frac{7\pi}{6}\right) = -\frac{1}{2}\) and \(\csc\left(\frac{7\pi}{6}\right) = -2\).<br />## Step 4: Calculate Cosine and Secant<br />### For the reference angle \( \frac{\pi}{6} \), \(\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}\). Thus, \(\cos\left(\frac{7\pi}{6}\right) = -\frac{\sqrt{3}}{2}\) and \(\sec\left(\frac{7\pi}{6}\right) = -\frac{2}{\sqrt{3}}\).<br />## Step 5: Calculate Tangent and Cotangent<br />### For the reference angle \( \frac{\pi}{6} \), \(\tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}}\). Thus, \(\tan\left(\frac{7\pi}{6}\right) = \frac{1}{\sqrt{3}}\) and \(\cot\left(\frac{7\pi}{6}\right) = \sqrt{3}\).