If `3costheta + 4sin theta = A sin (theta +alpha)`, then values of A and `alpha` are

Options

A. `5*cos^(-1)((3)/(5))`
B. `5, sin^(-1)((3)/(5))`
C. `7, sin^(-1)((4)/(5))`
D. None of these

Explanation

To solve the equation \(3 \cos \theta + 4 \sin \theta = A \sin(\theta + \alpha)\), we will follow these steps:

### Step 1: Rewrite the Right-Hand Side
Using the sine addition formula, we can rewrite the right-hand side:
\[ A \sin(\theta + \alpha) = A (\sin \theta \cos \alpha + \cos \theta \sin \alpha) \]
This gives us:
\[ A \sin(\theta + \alpha) = A \cos \alpha \sin \theta + A \sin \alpha \cos \theta \]

### Step 2: Compare Coefficients
Now, we can compare the coefficients of \(\sin \theta\) and \(\cos \theta\) from both sides of the equation:
- From \(3 \cos \theta + 4 \sin \theta\), we have:
- Coefficient of \(\sin \theta\) = 4
- Coefficient of \(\cos \theta\) = 3

From \(A \cos \alpha \sin \theta + A \sin \alpha \cos \theta\), we have:
- Coefficient of \(\sin \theta\) = \(A \cos \alpha\)
- Coefficient of \(\cos \theta\) = \(A \sin \alpha\)

### Step 3: Set Up Equations
From the comparison, we can set up the following equations:
1. \(A \sin \alpha = 3\) (1)
2. \(A \cos \alpha = 4\) (2)

### Step 4: Square and Add the Equations
To find \(A\), we can square both equations and add them:
\[ (A \sin \alpha)^2 + (A \cos \alpha)^2 = 3^2 + 4^2 \]
This simplifies to:
\[ A^2 (\sin^2 \alpha + \cos^2 \alpha) = 9 + 16 \]
Since \(\sin^2 \alpha + \cos^2 \alpha = 1\), we have:
\[ A^2 = 25 \]
Thus, taking the square root:
\[ A = 5 \]

### Step 5: Find \(\alpha\)
Now, we can substitute \(A = 5\) back into either equation (1) or (2) to find \(\alpha\). Using equation (1):
\[ 5 \sin \alpha = 3 \implies \sin \alpha = \frac{3}{5} \]
To find \(\alpha\):
\[ \alpha = \sin^{-1}\left(\frac{3}{5}\right) \]

### Final Result
Thus, the values of \(A\) and \(\alpha\) are:
\[ A = 5, \quad \alpha = \sin^{-1}\left(\frac{3}{5}\right) \]