Give a right angle triangle ABC with right angled at C. `angleC=90^(@)` and a,b,c are lengths of corresponding sides. `(b gt a)`.
sin2A will be equal to

Options

A. `(2bc)/a^(2)`
B. `(b^(2)-c^(2))/(c^(2))`
C. `(2ab)/c^(2)`
D. `(b^(2)-c^(2))/(2bc)`

Answer

To find the value of \( \sin 2A \) in a right triangle ABC with a right angle at C, we can follow these steps:

### Step 1: Understand the triangle
Given a right triangle ABC where angle C is \( 90^\circ \), we have:
- Side opposite angle A is \( a \)
- Side opposite angle B is \( b \)
- Hypotenuse (opposite angle C) is \( c \)

### Step 2: Use the Pythagorean theorem
According to the Pythagorean theorem:
\[ c^2 = a^2 + b^2 \]
From this, we can express \( b^2 \) as:
\[ b^2 = c^2 - a^2 \]

### Step 3: Use the sine definition
Using the sine definition for angle A:
\[ \sin A = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{c} \]

### Step 4: Use the double angle formula
To find \( \sin 2A \), we use the double angle formula:
\[ \sin 2A = 2 \sin A \cos A \]

### Step 5: Find \( \cos A \)
Using the cosine definition for angle A:
\[ \cos A = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{b}{c} \]

### Step 6: Substitute values into the double angle formula
Now substituting \( \sin A \) and \( \cos A \) into the double angle formula:
\[ \sin 2A = 2 \left(\frac{a}{c}\right) \left(\frac{b}{c}\right) \]
This simplifies to:
\[ \sin 2A = \frac{2ab}{c^2} \]

### Step 7: Substitute \( c^2 \) from the Pythagorean theorem
From the Pythagorean theorem, we know \( c^2 = a^2 + b^2 \). Thus, we can rewrite:
\[ \sin 2A = \frac{2ab}{a^2 + b^2} \]

### Final Answer
The value of \( \sin 2A \) is:
\[ \sin 2A = \frac{2ab}{a^2 + b^2} \]

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