If `sintheta= a` for e xactly one value of `theta in [0,(7pi)/3]`, then 10, ,then the value of a is

Options

A. `sqrt(3)/2`
B. 1
C. Zero
D. `-1`

Answer

To solve the problem, we need to find the value of \( a \) such that \( \sin \theta = a \) has exactly one solution for \( \theta \) in the interval \( [0, \frac{7\pi}{3}] \).

### Step-by-Step Solution:

1. **Understand the Range of Sine Function**:
The sine function oscillates between -1 and 1. Therefore, for \( \sin \theta = a \) to have solutions, \( a \) must be in the interval \([-1, 1]\).

**Hint**: Recall that the sine function has a maximum value of 1 and a minimum value of -1.

2. **Identify the Behavior of Sine Function**:
The sine function is periodic with a period of \( 2\pi \). In the interval \( [0, \frac{7\pi}{3}] \), we can determine how many times the sine function crosses a horizontal line at \( y = a \).

**Hint**: Sketch the sine wave from \( 0 \) to \( 2\pi \) and observe its behavior in the interval up to \( \frac{7\pi}{3} \).

3. **Determine the Values of \( a \)**:
- If \( a = 1 \): The equation \( \sin \theta = 1 \) has exactly one solution at \( \theta = \frac{\pi}{2} \).
- If \( a = -1 \): The equation \( \sin \theta = -1 \) has exactly one solution at \( \theta = \frac{3\pi}{2} \).
- If \( a = 0 \): The equation \( \sin \theta = 0 \) has three solutions: \( \theta = 0, \pi, 2\pi \).
- If \( a \) is any value between -1 and 1 (excluding -1 and 1), there will be two solutions in the interval.

**Hint**: Analyze how many intersections occur for different values of \( a \) within the range of the sine function.

4. **Conclusion**:
The only values of \( a \) for which \( \sin \theta = a \) has exactly one solution in the interval \( [0, \frac{7\pi}{3}] \) are \( a = 1 \) and \( a = -1 \).

**Final Answer**: The values of \( a \) are \( 1 \) and \( -1 \).