Calculate the value of A, if :
`cos3A.(2sin2A-1)=0`

Answer

To solve the equation \( \cos 3A (2 \sin^2 A - 1) = 0 \), we can break it down into two separate cases since the product of two factors is zero if at least one of the factors is zero.

### Step 1: Set each factor to zero
We have two factors:
1. \( \cos 3A = 0 \)
2. \( 2 \sin^2 A - 1 = 0 \)

### Step 2: Solve the first factor \( \cos 3A = 0 \)
The cosine function is zero at odd multiples of \( \frac{\pi}{2} \):
\[ 3A = \frac{\pi}{2} + n\pi \quad \text{for } n \in \mathbb{Z} \]
Dividing by 3 gives:
\[ A = \frac{\pi}{6} + \frac{n\pi}{3} \]
For \( n = 0 \):
\[ A = \frac{\pi}{6} \quad \text{(which is } 30^\circ\text{)} \]

### Step 3: Solve the second factor \( 2 \sin^2 A - 1 = 0 \)
Rearranging gives:
\[ 2 \sin^2 A = 1 \quad \Rightarrow \quad \sin^2 A = \frac{1}{2} \]
Taking the square root:
\[ \sin A = \frac{1}{\sqrt{2}} \quad \text{or} \quad \sin A = -\frac{1}{\sqrt{2}} \]
This corresponds to:
\[ A = \frac{\pi}{4} \quad \text{(which is } 45^\circ\text{)} \quad \text{and} \quad A = \frac{7\pi}{4} \quad \text{(which is } 315^\circ\text{)} \]

### Step 4: Compile the solutions
The solutions for \( A \) are:
1. \( A = 30^\circ \)
2. \( A = 45^\circ \)
3. \( A = 315^\circ \)

### Final Answer
Thus, the values of \( A \) that satisfy the equation are:
\[ A = 30^\circ, 45^\circ, 315^\circ \]

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