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