The largest value of `sin theta cos theta ` is

Options

A. 1
B. `1/2`
C. `(1)/(sqrt2)`
D. `(sqrt3)/(2)`

Answer

To find the largest value of \( \sin \theta \cos \theta \), we can follow these steps:

### Step 1: Use the double angle identity
We know that:
\[ \sin 2\theta = 2 \sin \theta \cos \theta \]
Thus, we can express \( \sin \theta \cos \theta \) in terms of \( \sin 2\theta \):
\[ \sin \theta \cos \theta = \frac{1}{2} \sin 2\theta \]

### Step 2: Determine the range of \( \sin 2\theta \)
The sine function, \( \sin x \), has a maximum value of 1. Therefore:
\[ -1 \leq \sin 2\theta \leq 1 \]

### Step 3: Find the maximum value of \( \sin \theta \cos \theta \)
Since \( \sin \theta \cos \theta = \frac{1}{2} \sin 2\theta \), we can find the maximum value of \( \sin \theta \cos \theta \) by substituting the maximum value of \( \sin 2\theta \):
\[ \text{Maximum value of } \sin \theta \cos \theta = \frac{1}{2} \times 1 = \frac{1}{2} \]

### Conclusion
Thus, the largest value of \( \sin \theta \cos \theta \) is:
\[ \frac{1}{2} \]

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