If `"sin" x = lambda` has exactly one solution in `[0, 9 pi//4]` then the number of values of `lambda`, is

Options

A. 0
B. 1
C. 2
D. 3

Explanation

To solve the problem, we need to determine the number of values of \(\lambda\) such that the equation \(\sin x = \lambda\) has exactly one solution in the interval \([0, \frac{9\pi}{4}]\).

### Step 1: Understand the behavior of the sine function
The function \(\sin x\) oscillates between -1 and 1. It has a period of \(2\pi\). In the interval \([0, \frac{9\pi}{4}]\), we can find how many complete cycles of \(\sin x\) fit.

### Step 2: Determine the number of cycles in the interval
The length of the interval \([0, \frac{9\pi}{4}]\) is \(\frac{9\pi}{4} - 0 = \frac{9\pi}{4}\). The period of \(\sin x\) is \(2\pi\).

To find how many complete cycles fit in this interval:
\[ \text{Number of cycles} = \frac{\frac{9\pi}{4}}{2\pi} = \frac{9}{8} \]
This means there are 1 complete cycle and a bit more (specifically, \(\frac{1}{8}\) of a cycle).

### Step 3: Sketch the graph of \(\sin x\)
The graph of \(\sin x\) will start at 0, reach 1 at \(\frac{\pi}{2}\), go back to 0 at \(\pi\), reach -1 at \(\frac{3\pi}{2}\), and return to 0 at \(2\pi\). After \(2\pi\), it will continue to oscillate.

In the interval \([0, \frac{9\pi}{4}]\), the sine function will complete one full cycle and then continue to \(\frac{9\pi}{4}\), which is \(\frac{1}{8}\) of the next cycle.

### Step 4: Identify the maximum and minimum values
The maximum value of \(\sin x\) is 1, and the minimum value is -1.

### Step 5: Determine where \(\sin x = \lambda\) has exactly one solution
For \(\sin x = \lambda\) to have exactly one solution, \(\lambda\) must be equal to the maximum or minimum value of \(\sin x\) at the endpoints of the interval:

1. If \(\lambda = 1\), the horizontal line \(y = 1\) will intersect the sine curve at exactly one point (the peak of the sine wave).
2. If \(\lambda = -1\), the horizontal line \(y = -1\) will also intersect the sine curve at exactly one point (the trough of the sine wave).

### Conclusion
Thus, the values of \(\lambda\) for which \(\sin x = \lambda\) has exactly one solution in the interval \([0, \frac{9\pi}{4}]\) are \(\lambda = 1\) and \(\lambda = -1\).

Therefore, the number of values of \(\lambda\) is **2**.