If `cotA=(7)/(24)`, then the value of `sin A` is :

Options

A. `(24)/(7)`
B. `(24)/(25)`
C. `(25)/(24)`
D. `(7)/(25)`.

Answer

To find the value of \( \sin A \) given that \( \cot A = \frac{7}{24} \), we can follow these steps:

### Step 1: Understand the relationship of cotangent
We know that:
\[ \cot A = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{b}{p} \]
From the problem, we have:
\[ \cot A = \frac{7}{24} \]
This means:
- Adjacent side (base, \( b \)) = 7
- Opposite side (perpendicular, \( p \)) = 24

### Step 2: Use the Pythagorean theorem to find the hypotenuse
Using the Pythagorean theorem:
\[ h^2 = b^2 + p^2 \]
Substituting the values of \( b \) and \( p \):
\[ h^2 = 7^2 + 24^2 \]
Calculating the squares:
\[ h^2 = 49 + 576 = 625 \]
Taking the square root:
\[ h = \sqrt{625} = 25 \]

### Step 3: Calculate \( \sin A \)
Now that we have the lengths of all sides, we can find \( \sin A \):
\[ \sin A = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{p}{h} \]
Substituting the values:
\[ \sin A = \frac{24}{25} \]

### Final Answer
Thus, the value of \( \sin A \) is:
\[ \sin A = \frac{24}{25} \]

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