If theta is an acute angle of a right triangle and if cosecant theta equals 3​, then find the values of the remaining five trigonometric functions for angle theta.

To find the values of the remaining five trigonometric functions for an acute angle \(\theta\) in a right triangle, given that \(\csc \theta = 3\), follow these steps:

### Step 1: Understand the given value
Given:
\[
\csc \theta = 3
\]
Since \(\csc \theta = \frac{1}{\sin \theta}\), we have:
\[
\sin \theta = \frac{1}{\csc \theta} = \frac{1}{3}
\]

### Step 2: Calculate \(\cos \theta\)
Using the Pythagorean identity:
\[
\sin^2 \theta + \cos^2 \theta = 1
\]
Substitute \(\sin \theta = \frac{1}{3}\):
\[
\left(\frac{1}{3}\right)^2 + \cos^2 \theta = 1
\]
\[
\frac{1}{9} + \cos^2 \theta = 1
\]
\[
\cos^2 \theta = 1 - \frac{1}{9} = \frac{9}{9} - \frac{1}{9} = \frac{8}{9}
\]
\[
\cos \theta = \sqrt{\frac{8}{9}} = \frac{\sqrt{8}}{3} = \frac{2\sqrt{2}}{3}
\]

### Step 3: Calculate \(\tan \theta\)
\[
\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{1}{3}}{\frac{2\sqrt{2}}{3}} = \frac{1}{2\sqrt{2}} = \frac{\sqrt{2}}{4}
\]

### Step 4: Calculate \(\cot \theta\)
\[
\cot \theta = \frac{1}{\tan \theta} = \frac{1}{\frac{\sqrt{2}}{4}} = \frac{4}{\sqrt{2}} = 2\sqrt{2}
\]

### Step 5: Calculate \(\sec \theta\)
\[
\sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{2\sqrt{2}}{3}} = \frac{3}{2\sqrt{2}} = \frac{3\sqrt{2}}{4}
\]

### Step 6: Summarize the values of all six trigonometric functions
\[
\sin \theta = \frac{1}{3}
\]
\[
\cos \theta = \frac{2\sqrt{2}}{3}
\]
\[
\tan \theta = \frac{\sqrt{2}}{4}
\]
\[
\cot \theta = 2\sqrt{2}
\]
\[
\sec \theta = \frac{3\sqrt{2}}{4}
\]
\[
\csc \theta = 3
\]

Do you want details or have any questions?

### Related Questions

1. How do you derive the Pythagorean identity used in trigonometry?
2. Can you explain why the value of \(\csc \theta\) must always be greater than or equal to 1 for acute angles?
3. What is the significance of the \(\sec \theta\) and \(\csc \theta\) values in a right triangle?
4. How does knowing one trigonometric function help in finding the others in a right triangle?
5. Why is \(\theta\) an acute angle in this context, and how does that affect the values of the trigonometric functions?
6. What are the practical applications of these trigonometric functions in real-world problems?
7. How can the trigonometric ratios be used to solve problems involving right triangles?
8. Can you demonstrate how to verify these trigonometric values using a calculator?

### Tip

Remember that the reciprocal relationships among trigonometric functions are crucial for solving many problems quickly and efficiently.

Learn how to find the values of all six trigonometric functions for an acute angle \(\theta\) in a right triangle, given that \(\csc \theta = 3\). Detailed step-by-step solution covering trigonometric identities and formulas. Suitable for high school students studying trigonometry.

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