What is the maximum value of sin3`theta` cos 2`theta` + cos 3`theta` sin 2`theta`?

Options

A. 1
B. 2
C. 4
D. 10

Answer

To find the maximum value of the expression \( \sin 3\theta \cos 2\theta + \cos 3\theta \sin 2\theta \), we can use the sine addition formula.

### Step-by-Step Solution:

1. **Identify the Expression**:
We are given the expression \( \sin 3\theta \cos 2\theta + \cos 3\theta \sin 2\theta \).

2. **Apply the Sine Addition Formula**:
Recall that the sine addition formula states:
\[ \sin(a + b) = \sin a \cos b + \cos a \sin b \]
Here, we can let \( a = 3\theta \) and \( b = 2\theta \). Thus, we can rewrite the expression as:
\[ \sin(3\theta + 2\theta) = \sin(5\theta) \]

3. **Determine the Maximum Value of Sine**:
The sine function, \( \sin x \), has a maximum value of 1 for any angle \( x \). Therefore, the maximum value of \( \sin(5\theta) \) is:
\[ \max(\sin(5\theta)) = 1 \]

4. **Conclusion**:
Hence, the maximum value of the original expression \( \sin 3\theta \cos 2\theta + \cos 3\theta \sin 2\theta \) is:
\[ \boxed{1} \]