Understanding the Equation
The equation given is:
**A sin(theta) + B cos(theta) = A sin(theta + phi)**
This can be interpreted as a linear combination of sine and cosine functions equating to a sine function with a phase shift.

Identifying A and φ
To find the values of **A** and **φ**, we can use the identity for the sine of a sum:
**sin(theta + phi) = sin(theta)cos(phi) + cos(theta)sin(phi)**
This allows us to express the right side of the equation in terms of sine and cosine.

Comparing Coefficients
We can rewrite the equation as:
**A sin(theta) + B cos(theta) = A (sin(theta)cos(phi) + cos(theta)sin(phi))**
Now, equate coefficients from both sides:
- Coefficient of sin(theta):
**A = A cos(phi)**
- Coefficient of cos(theta):
**B = A sin(phi)**

Finding A and φ
From the first equation, we can derive:
**A (1 - cos(phi)) = 0**
Hence, **A ≠ 0** implies **cos(phi) = 1**, which leads to **φ = 0**.
From the second equation:
**B = A sin(phi)**
Substituting φ = 0 gives **B = 0**.

Final Values
The determined values are:
- **A = A** (any non-zero value)
- **φ = 0**
This means the original equation holds when φ is zero, and A remains as a constant multiplier of the sine function.