A function value and a quadrant are given. Find the other five function values. Give exact answers. $\cot \theta=4$, Quadrant III

Step 1 ：Given that \(\cot \theta = 4\) and \(\theta\) is in the third quadrant, we know that both sine and cosine are negative in this quadrant.

Step 2 ：Since cotangent is the reciprocal of tangent, we can find the value of tangent by taking the reciprocal of the cotangent value. So, \(\tan \theta = \frac{1}{\cot \theta} = \frac{1}{4} = 0.25\).

Step 3 ：We can use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) to find the values of sine and cosine. Solving for \(\sin \theta\), we get \(\sin \theta = \sqrt{1 - \cos^2 \theta}\). Since \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), we can substitute \(\tan \theta\) and \(\sin \theta\) into the equation to get \(\cos \theta = -\sqrt{1 - (4 \tan \theta)^2} = -0.2425\) and \(\sin \theta = -\sqrt{1 - \cos^2 \theta} = -0.9701\).

Step 4 ：The values of cosecant, secant, and cotangent can be found by taking the reciprocals of sine, cosine, and tangent respectively. So, \(\csc \theta = \frac{1}{\sin \theta} = -1.0308\), \(\sec \theta = \frac{1}{\cos \theta} = -4.1231\), and \(\cot \theta = 4\).

Step 5 ：\(\boxed{\text{Final Answer: The other five function values are } \sin \theta = -0.9701, \cos \theta = -0.2425, \tan \theta = 0.25, \csc \theta = -1.0308, \sec \theta = -4.1231}\)