If `sin2A=(4)/(5)`, then what is the value of `tanA(0leAle(pi)/(4))`?

Options

A. 1
B. `-1`
C. `(1)/(2)`
D. 2

Explanation

To find the value of \( \tan A \) given that \( \sin 2A = \frac{4}{5} \) and \( 0 \leq A \leq \frac{\pi}{4} \), we can follow these steps:

### Step 1: Use the double angle identity for sine
We know that:
\[ \sin 2A = 2 \sin A \cos A \]
But we can also express \( \sin 2A \) in terms of \( \tan A \):
\[ \sin 2A = \frac{2 \tan A}{1 + \tan^2 A} \]
Setting this equal to the given value:
\[ \frac{2 \tan A}{1 + \tan^2 A} = \frac{4}{5} \]

### Step 2: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 5 \cdot 2 \tan A = 4(1 + \tan^2 A) \]
This simplifies to:
\[ 10 \tan A = 4 + 4 \tan^2 A \]

### Step 3: Rearrange the equation into standard quadratic form
Rearranging the equation results in:
\[ 4 \tan^2 A - 10 \tan A + 4 = 0 \]

### Step 4: Solve the quadratic equation
We can use the quadratic formula \( \tan A = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 4, b = -10, c = 4 \):
\[ \tan A = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 4 \cdot 4}}{2 \cdot 4} \]
Calculating the discriminant:
\[ \tan A = \frac{10 \pm \sqrt{100 - 64}}{8} = \frac{10 \pm \sqrt{36}}{8} = \frac{10 \pm 6}{8} \]
This gives us two possible solutions:
\[ \tan A = \frac{16}{8} = 2 \quad \text{and} \quad \tan A = \frac{4}{8} = \frac{1}{2} \]

### Step 5: Determine the valid solution
Given the constraint \( 0 \leq A \leq \frac{\pi}{4} \), we know that \( \tan A \) must be non-negative and less than or equal to 1. Therefore, we discard \( \tan A = 2 \) and accept:
\[ \tan A = \frac{1}{2} \]

### Final Answer
Thus, the value of \( \tan A \) is:
\[ \boxed{\frac{1}{2}} \]