Given that 

sine thetasinθequals=StartFraction 3 Over StartRoot 14 EndRoot EndFraction314 

and 

cosine thetacosθequals=StartFraction StartRoot 5 EndRoot Over StartRoot 14 EndRoot EndFraction514 

find the values of the remaining four trigonometric functions using identities.

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.

Learn how to find the values of trigonometric functions such as sine, cosine, tangent, cotangent, secant, and cosecant using given trigonometric identities and formulas. Detailed step-by-step solution provided.

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