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

$\sin \theta = \frac{1}{3}$ $\cos \theta = \frac{2\sqrt{2}}{3}$ $\tan \theta = \frac{\sqrt{2}}{4}$ $\cot \theta = 2\sqrt{2}$ $\sec \theta = \frac{3\sqrt{2}}{4}$ $\csc \theta = 3$

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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.