If in `DeltaABC, A =90^(@)` and c, sin B and cos B are rational number, then

Options

A. a is rational
B. a is irrational
C. b is rational
D. b is irational

Answer

To solve the problem step by step, let's analyze the given information about triangle ABC where angle A is 90 degrees, and the values of side c, sin B, and cos B are rational numbers.

### Step 1: Understand the Triangle and Given Information
In triangle ABC:
- Angle A = 90 degrees
- Side opposite angle A is denoted as 'a'
- Side opposite angle B is denoted as 'b'
- Side opposite angle C is denoted as 'c'

Given:
- c (the length of side opposite angle C) is a rational number.
- sin B and cos B are rational numbers.

### Step 2: Express sin B and cos B
Using the definitions of sine and cosine for angle B in a right triangle:
- \( \sin B = \frac{b}{a} \)
- \( \cos B = \frac{c}{a} \)

### Step 3: Analyze the Rationality of sin B and cos B
Since both sin B and cos B are given to be rational, we can conclude:
- \( \frac{b}{a} \) is rational
- \( \frac{c}{a} \) is rational

### Step 4: Implications of Rational Values
Since \( \frac{b}{a} \) is rational, it implies that both b and a must be rational numbers (as long as a is not zero). Similarly, since \( \frac{c}{a} \) is rational and c is given to be rational, it follows that a must also be rational.

### Step 5: Conclude the Rationality of All Sides
From the previous steps:
- Since c is rational and \( \frac{c}{a} \) is rational, a must be rational.
- Since \( \frac{b}{a} \) is rational and a is rational, b must also be rational.

Thus, we conclude that:
- a, b, and c are all rational numbers.

### Final Conclusion
Since all sides of triangle ABC (a, b, c) are rational, we can confirm that:
- Option A (c is rational) is correct.
- Option B (sin B is rational) is correct.
- Option C (cos B is rational) is correct.