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To solve for \(\sin A\), \(\cos A\), and \(\tan A\) in the given right triangle, we first need to find the hypotenuse \(c\) using the Pythagorean theorem. The side lengths \(a\) and \(b\) are both given as \(14 \sqrt{7}\).

### Step 1: Find the length of the hypotenuse \(c\)

By the Pythagorean theorem:

\[
c = \sqrt{a^2 + b^2}
\]

Substitute the values of \(a\) and \(b\):

\[
c = \sqrt{(14 \sqrt{7})^2 + (14 \sqrt{7})^2}
\]

Calculate \( (14 \sqrt{7})^2 \):

\[
(14 \sqrt{7})^2 = 14^2 \cdot (\sqrt{7})^2 = 196 \cdot 7 = 1372
\]

So,

\[
c = \sqrt{1372 + 1372} = \sqrt{2744} = 52
\]

### Step 2: Calculate \(\sin A\), \(\cos A\), and \(\tan A\)

Now we know:
- \(a = 14 \sqrt{7}\)
- \(b = 14 \sqrt{7}\)
- \(c = 52\)

#### \(\sin A\)

\[
\sin A = \frac{\text{opposite side}}{\text{hypotenuse}} = \frac{b}{c} = \frac{14 \sqrt{7}}{52}
\]

Simplify the fraction:

\[
\sin A = \frac{14 \sqrt{7}}{52} = \frac{\sqrt{7}}{4}
\]

#### \(\cos A\)

\[
\cos A = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{a}{c} = \frac{14 \sqrt{7}}{52}
\]

Simplify the fraction:

\[
\cos A = \frac{14 \sqrt{7}}{52} = \frac{\sqrt{7}}{4}
\]

#### \(\tan A\)

\[
\tan A = \frac{\text{opposite side}}{\text{adjacent side}} = \frac{b}{a} = \frac{14 \sqrt{7}}{14 \sqrt{7}} = 1
\]

### Final Answers

- \(\sin A = \frac{\sqrt{7}}{4}\)
- \(\cos A = \frac{\sqrt{7}}{4}\)
- \(\tan A = 1\)

Would you like to ask more questions or dive deeper into any part of the solution? Here are some related questions:

1. What is the definition of the trigonometric ratios in a right triangle?
2. How does the Pythagorean theorem work for finding the hypotenuse?
3. What are the reciprocal trigonometric functions, like cosecant, secant, and cotangent?
4. Can you explain the relationship between \(\sin\), \(\cos\), and \(\tan\) for different angles?
5. What happens to the values of \(\sin A\), \(\cos A\), and \(\tan A\) as angle \(A\) changes?

**Tip:** Remember that in a right triangle, the hypotenuse is always the longest side.

Suppose a = b = 14√7. Find sin A, cos A, and tan A for the right triangle with sides a, b, and hypotenuse c.

Learn how to solve for sin A, cos A, and tan A in a right triangle where both legs are a = b = 14√7. This solution covers the Pythagorean theorem and trigonometric ratios, including exact simplifications. Suitable for students in Grades 9-12.

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