It seems like there might be a typo in the cosine value you provided, as the cosine of an angle cannot be greater than 1 in absolute value. Assuming you meant the cosine to be \(-\frac{16}{\sqrt{17}}\) (since \(-16\sqrt{17}\) is not feasible), I will proceed with that assumption.
Given: \[ \cos(\theta) = -\frac{16}{\sqrt{17}} \]
Step 1: Find \(\sin(\theta)\) using the Pythagorean identity
The Pythagorean identity tells us that: \[ \sin^2(\theta) + \cos^2(\theta) = 1 \] Substituting the value of \(\cos(\theta)\): \[ \sin^2(\theta) + \left(-\frac{16}{\sqrt{17}}\right)^2 = 1 \] Calculating \(\cos^2(\theta)\): \[ \sin^2(\theta) + \frac{256}{17} = 1 \] Rearranging gives: \[ \sin^2(\theta) = 1 - \frac{256}{17} = \frac{17}{17} - \frac{256}{17} = \frac{-239}{17} \] This is incorrect, as the sine squared cannot be negative. Thus the assumption about the cosine might be completely off.
Consider an alternate approach
Visitors often provide ratios instead of exact values. If you confirm or provide the right ratio for \(\cos(\theta)\), for example, \(-\frac{3}{5}\) or any proper trigonometric ratio, then we can proceed with calculations based solely on that known quantity.
Step 2: Determine \(\tan(\theta)\)
The tangent identity states: \[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \] So after finding \(\sin(\theta)\), you can find \(\tan(\theta)\) accordingly.
Please clarify the cosine value so we can continue with accurate calculations!