The total number of ordered pairs (x, y) satisfying `|y|=cosx and y=sin^(-1)(sinx)`, where `x in [-2pi, 3pi]` is equal to :

Options

A. 2
B. 4
C. 5
D. 6

Answer

To solve the problem of finding the total number of ordered pairs \((x, y)\) satisfying the equations \(|y| = \cos x\) and \(y = \sin^{-1}(\sin x)\) for \(x \in [-2\pi, 3\pi]\), we can follow these steps:

### Step 1: Analyze the equation \(y = \sin^{-1}(\sin x)\)

The function \(y = \sin^{-1}(\sin x)\) gives the principal value of the sine function, which means it will return values in the range \([- \frac{\pi}{2}, \frac{\pi}{2}]\). However, the output will depend on the value of \(x\):

- For \(x \in [2n\pi - \frac{\pi}{2}, 2n\pi + \frac{\pi}{2}]\), \(y = x\)
- For \(x \in [2n\pi + \frac{\pi}{2}, 2n\pi + \frac{3\pi}{2}]\), \(y = \pi - x\)
- For \(x \in [2n\pi - \frac{3\pi}{2}, 2n\pi - \frac{\pi}{2}]\), \(y = -x\)

### Step 2: Analyze the equation \(|y| = \cos x\)

The equation \(|y| = \cos x\) means that \(y\) can take two values: \(y = \cos x\) and \(y = -\cos x\).

### Step 3: Find intersections of the graphs

We need to find the points of intersection of the graphs of \(y = \cos x\) and \(y = \sin^{-1}(\sin x)\) within the given range of \(x\).

1. **For \(y = \cos x\)**:
- The cosine function oscillates between -1 and 1, with a period of \(2\pi\).

2. **For \(y = \sin^{-1}(\sin x)\)**:
- The function oscillates between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\).

### Step 4: Determine the range of \(x\)

The range of \(x\) is \([-2\pi, 3\pi]\). We will analyze the behavior of both functions in this interval.

### Step 5: Count the intersections

We will sketch the graphs of \(|y| = \cos x\) and \(y = \sin^{-1}(\sin x)\) to find the points of intersection.

1. From \(-2\pi\) to \(0\):
- The cosine function will have intersections with the sine inverse function.

2. From \(0\) to \(3\pi\):
- The cosine function continues to oscillate, and we will find additional intersections.

### Step 6: Conclusion

After analyzing the graphs and counting the intersections, we find that there are a total of **5 points of intersection** where the two functions meet.

Thus, the total number of ordered pairs \((x, y)\) satisfying the given equations is **5**.

### Final Answer:
The total number of ordered pairs \((x, y)\) is **5**.

---