Given the following information evaluate the other five functions.

cosecant theta equals negative five thirdscscθ=−53 

and StartFraction 3 pi Over 2 EndFraction less than theta less than 2 pi3π2<θ<2π

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Part 1

sine theta equalssinθ=enter your response here
cos theta
tan theta
cot theta
csc theta

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!

Given the following information evaluate the other five functions: cosecant theta equals negative five thirds (cscθ = -5/3) and 3π/2 < θ < 2π. Part 1: sin(θ), cos(θ), tan(θ), cot(θ), csc(θ)

Learn how to evaluate trigonometric functions when given csc(θ) = -5/3 and θ in the fourth quadrant (3π/2 < θ < 2π). This problem explores the use of trigonometric identities, including the Pythagorean identity, and how to determine the signs of trigonometric functions based on the quadrant. Suitable for students in Grades 10-12.

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