Given: $\sin \theta = \frac{3}{\sqrt{14}}$ $\cos \theta = \frac{\sqrt{5}}{\sqrt{14}}$

We need to find the values of the remaining four trigonometric functions: $\tan \theta$, $\cot \theta$, $\sec \theta$, and $\csc \theta$.

Step 1: Find $\tan \theta$

$\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{3}{\sqrt{14}}}{\frac{\sqrt{5}}{\sqrt{14}}} = \frac{3}{\sqrt{5}} = \frac{3\sqrt{5}}{5}$

Step 2: Find $\cot \theta$

$\cot \theta = \frac{1}{\tan \theta} = \frac{1}{\frac{3\sqrt{5}}{5}} = \frac{5}{3\sqrt{5}} = \frac{\sqrt{5}}{3}$

Step 3: Find $\sec \theta$

$\sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{\sqrt{5}}{\sqrt{14}}} = \frac{\sqrt{14}}{\sqrt{5}} = \sqrt{\frac{14}{5}}$

Step 4: Find $\csc \theta$

$\csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{3}{\sqrt{14}}} = \frac{\sqrt{14}}{3}$

Summary of all six trigonometric functions:

$\sin \theta = \frac{3}{\sqrt{14}}$ $\cos \theta = \frac{\sqrt{5}}{\sqrt{14}}$ $\tan \theta = \frac{3\sqrt{5}}{5}$ $\cot \theta = \frac{\sqrt{5}}{3}$ $\sec \theta = \sqrt{\frac{14}{5}}$ $\csc \theta = \frac{\sqrt{14}}{3}$

Do you want details or have any questions?

Related Questions

1. How do you rationalize the denominator of a trigonometric function?
2. What are the relationships between the primary and reciprocal trigonometric functions?
3. Can you derive $\tan \theta$ from $\sin \theta$ and $\cos \theta$?
4. Why is it important to know all six trigonometric functions?
5. How can you verify these trigonometric values using a calculator?
6. What is the significance of the Pythagorean identity in solving trigonometric problems?
7. How are trigonometric functions applied in solving real-world problems?
8. Can you demonstrate solving a right triangle problem using these trigonometric functions?

Tip

When working with trigonometric functions, always ensure to simplify your expressions and rationalize the denominators for a more standard form of the answers.