- A and phi Calculation:
1. **Given Equation:**
$a \sin \theta + b \cos \theta = A \cos(\theta + \phi)$
2. **Identifying A and phi:**
By comparing the given equation with the standard form of cosine addition formula:
$A \cos(\theta + \phi) = A(\cos \theta \cos \phi - \sin \theta \sin \phi)$
We can see that:
$A = \sqrt{a^2 + b^2}$ and $\tan \phi = -\frac{b}{a}$
3. **Calculation:**
Using the above relations, we can calculate A and phi.
$A = \sqrt{a^2 + b^2}$
$\tan \phi = -\frac{b}{a}$
4. **Explanation:**
The coefficient of sin theta in the given equation is 'a' and in the standard form is $-A \sin \phi$.
Similarly, the coefficient of cos theta in the given equation is 'b' and in the standard form is $A \cos \phi$.
By comparing these coefficients, we can find the values of A and phi.
5. **Conclusion:**
Therefore, by comparing the given equation with the standard form of cosine addition formula, we can calculate the values of A and phi in terms of a and b.