$\sin \theta$ and $\cos \theta$ are given. Find the exact value of each of the four remaining trigonometric functions.
\[
\sin \theta=\frac{4}{5}, \cos \theta=-\frac{3}{5}
\]

Step 1 ：We are given that \(\sin \theta = \frac{4}{5}\) and \(\cos \theta = -\frac{3}{5}\).

Step 2 ：We can use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) to verify that these values are correct.

Step 3 ：Next, we use the definitions of the other trigonometric functions in terms of \(\sin \theta\) and \(\cos \theta\) to find their values.

Step 4 ：\(\tan \theta = \frac{\sin \theta}{\cos \theta} = -\frac{4}{3}\)

Step 5 ：\(\csc \theta = \frac{1}{\sin \theta} = \frac{5}{4}\)

Step 6 ：\(\sec \theta = \frac{1}{\cos \theta} = -\frac{5}{3}\)

Step 7 ：\(\cot \theta = \frac{1}{\tan \theta} = -\frac{3}{4}\)

Step 8 ：Final Answer: The exact values of the four remaining trigonometric functions are: \(\tan \theta = \boxed{-\frac{4}{3}}\), \(\csc \theta = \boxed{\frac{5}{4}}\), \(\sec \theta = \boxed{-\frac{5}{3}}\), \(\cot \theta = \boxed{-\frac{3}{4}}\)