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We are given a right triangle, and we need to find the trigonometric functions for angle \( A \). Here’s what we know from the figure:

- The side adjacent to angle \( A \) is 4.
- The side opposite angle \( A \) is 14.
- The hypotenuse is unknown, but we can calculate it using the Pythagorean theorem.

### Step 1: Calculate the Hypotenuse

Using the Pythagorean theorem:
\[
\text{Hypotenuse} = \sqrt{(\text{Adjacent side})^2 + (\text{Opposite side})^2}
\]
\[
\text{Hypotenuse} = \sqrt{4^2 + 14^2} = \sqrt{16 + 196} = \sqrt{212}
\]
\[
\text{Hypotenuse} = 2\sqrt{53}
\]

### Step 2: Find the Trigonometric Functions
Now, we can evaluate the six trigonometric functions for angle \( A \).

1. **Sine** (\(\sin A\)):
   \[
   \sin A = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{14}{2\sqrt{53}} = \frac{7}{\sqrt{53}}
   \]

2. **Cosine** (\(\cos A\)):
   \[
   \cos A = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{4}{2\sqrt{53}} = \frac{2}{\sqrt{53}}
   \]

3. **Tangent** (\(\tan A\)):
   \[
   \tan A = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{14}{4} = \frac{7}{2}
   \]

4. **Cosecant** (\(\csc A\)):
   \[
   \csc A = \frac{1}{\sin A} = \frac{\sqrt{53}}{7}
   \]

5. **Secant** (\(\sec A\)):
   \[
   \sec A = \frac{1}{\cos A} = \frac{\sqrt{53}}{2}
   \]

6. **Cotangent** (\(\cot A\)):
   \[
   \cot A = \frac{1}{\tan A} = \frac{2}{7}
   \]

### Final Answers:
- \(\sin A = \frac{7}{\sqrt{53}}\)
- \(\cos A = \frac{2}{\sqrt{53}}\)
- \(\tan A = \frac{7}{2}\)
- \(\csc A = \frac{\sqrt{53}}{7}\)
- \(\sec A = \frac{\sqrt{53}}{2}\)
- \(\cot A = \frac{2}{7}\)

Would you like more details on any of these calculations or further questions? 

Here are some related questions to expand your understanding:
1. How do you rationalize the denominator in the sine or cosine functions?
2. What is the significance of each trigonometric ratio in real-world applications?
3. How would the trigonometric values change if angle \( A \) were a different size?
4. Can the Pythagorean theorem be used to find sides in non-right triangles?
5. How does the unit circle relate to these trigonometric values?

**Tip**: Always remember to simplify fractions and rationalize the denominator for trigonometric expressions to give exact answers.

Use the triangle with side adjacent to angle A as 4 and side opposite angle A as 14 to evaluate each trigonometric function of angle A. Enter the exact answers.

This problem evaluates the trigonometric functions of angle A for a right triangle where the side opposite is 14 units and the adjacent side is 4 units. It involves using the Pythagorean theorem to find the hypotenuse and applying the sine, cosine, tangent, cosecant, secant, and cotangent formulas.

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