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: Understand the equations
We have two equations:
1. \(|y| = \cos x\)
2. \(y = \sin^{-1}(\sin x)\)

### Step 2: Analyze the first equation \(|y| = \cos x\)
The equation \(|y| = \cos x\) implies that:
- \(y = \cos x\) (for the positive case)
- \(y = -\cos x\) (for the negative case)

### Step 3: Analyze the second equation \(y = \sin^{-1}(\sin x)\)
The function \(y = \sin^{-1}(\sin x)\) gives:
- \(y = x\) when \(x\) is in the range \([- \frac{\pi}{2}, \frac{\pi}{2}]\)
- \(y = \pi - x\) when \(x\) is in the range \([\frac{\pi}{2}, \frac{3\pi}{2}]\)
- \(y = x - 2\pi\) when \(x\) is in the range \([\frac{3\pi}{2}, \frac{5\pi}{2}]\)
- \(y = -\pi - x\) when \(x\) is in the range \([- \frac{3\pi}{2}, -\frac{\pi}{2}]\)

### Step 4: Graph both equations
To find the points of intersection, we can graph both equations:
1. The graph of \(|y| = \cos x\) will oscillate between \(-1\) and \(1\) and will be symmetric about the x-axis.
2. The graph of \(y = \sin^{-1}(\sin x)\) will oscillate between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\) and will have a sawtooth pattern.

### Step 5: Determine the intervals for \(x\)
We need to consider the intervals for \(x\) from \([-2\pi, 3\pi]\). This range includes several periods of the cosine function.

### Step 6: Find points of intersection
By analyzing the graphs:
- For \(y = \cos x\), it intersects with \(y = \sin^{-1}(\sin x)\) at several points in the given range.
- Count the intersections for both \(y = \cos x\) and \(y = -\cos x\).

### Step 7: Count the total ordered pairs
After analyzing the graphs and counting the intersections, we find that there are a total of 5 points where the two graphs intersect.

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

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