Given the information $\csc(\theta) = -\frac{5}{3}$ and $\frac{3\pi}{2} < \theta < 2\pi$, let's evaluate the other trigonometric functions:

Step 1: Determine the Quadrant

The interval $\frac{3\pi}{2} < \theta < 2\pi$ places $\theta$ in the fourth quadrant.

• In the fourth quadrant:
  • $\sin(\theta)$ is negative.
  • $\cos(\theta)$ is positive.
  • $\tan(\theta)$ is negative.

Step 2: Evaluate $\sin(\theta)$

Since $\csc(\theta) = \frac{1}{\sin(\theta)}$, we have: $\sin(\theta) = \frac{1}{\csc(\theta)} = \frac{1}{-\frac{5}{3}} = -\frac{3}{5}$

Step 3: Evaluate $\cos(\theta)$

Use the Pythagorean identity $\sin^2(\theta) + \cos^2(\theta) = 1$: $\sin^2(\theta) = \left(-\frac{3}{5}\right)^2 = \frac{9}{25}$ $\cos^2(\theta) = 1 - \frac{9}{25} = \frac{16}{25}$ $\cos(\theta) = \sqrt{\frac{16}{25}} = \frac{4}{5}$ Since $\theta$ is in the fourth quadrant, $\cos(\theta)$ is positive: $\cos(\theta) = \frac{4}{5}$

Step 4: Evaluate $\tan(\theta)$

$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{-\frac{3}{5}}{\frac{4}{5}} = -\frac{3}{4}$

Step 5: Evaluate $\cot(\theta)$

$\cot(\theta) = \frac{1}{\tan(\theta)} = -\frac{4}{3}$

Step 6: Reevaluate $\csc(\theta)$ (Verification)

We already know: $\csc(\theta) = -\frac{5}{3}$

Summary of the Trigonometric Functions

$\sin(\theta) = -\frac{3}{5}$ $\cos(\theta) = \frac{4}{5}$ $\tan(\theta) = -\frac{3}{4}$ $\cot(\theta) = -\frac{4}{3}$ $\csc(\theta) = -\frac{5}{3}$

Would you like to delve into any specific detail or have further questions?

Here are 5 related questions:

1. How do the signs of trigonometric functions change in different quadrants?
2. What is the significance of the Pythagorean identity in trigonometry?
3. How can you verify the values of trigonometric functions in different quadrants?
4. What are the reciprocal identities in trigonometry?
5. How does the unit circle help in understanding trigonometric functions?

Tip: Always pay attention to the quadrant when determining the sign of trigonometric functions!