Find the number of ordered pairs of `(x, y)` satisfying the equation` y = |sinx|` and` y = cos^(-1)(cosx)`, where `x in [-π, π]`

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the number of ordered pairs \((x, y)\) satisfying the equations \(y = |\sin x|\) and \(y = \cos^{-1}(\cos x)\) for \(x \in [-\pi, \pi]\), we will analyze both functions graphically and algebraically.

### Step 1: Understand the functions

1. **Function \(y = |\sin x|\)**:
- The function \(|\sin x|\) oscillates between 0 and 1.
- It has a period of \(2\pi\) and is symmetric about the y-axis.
- Within the interval \([-π, π]\), it takes values from 0 to 1 at \(x = 0, \pm \frac{\pi}{2}, \pm \pi\).

2. **Function \(y = \cos^{-1}(\cos x)\)**:
- The function \(\cos^{-1}(\cos x)\) is defined as:
- \(y = x\) for \(x \in [0, \pi]\)
- \(y = -x\) for \(x \in [-\pi, 0]\)
- Thus, within the interval \([-π, π]\), it also takes values from 0 to \(\pi\).

### Step 2: Graph the functions

- **Graph of \(y = |\sin x|\)**:
- Peaks at \((-\frac{\pi}{2}, 1)\), \((\frac{\pi}{2}, 1)\) and touches the x-axis at \(x = -\pi, 0, \pi\).

- **Graph of \(y = \cos^{-1}(\cos x)\)**:
- Starts at \((-\pi, 0)\), increases to \((0, \frac{\pi}{2})\), and then to \((\pi, \pi)\).

### Step 3: Find intersections

To find the number of ordered pairs \((x, y)\), we need to find the points where the two graphs intersect.

1. **Intersection points**:
- For \(x \in [-\pi, 0]\):
- \(y = |\sin x| = -\sin x\) (since \(\sin x\) is negative in this interval).
- The equation becomes \(-\sin x = -x\) or \(\sin x = x\).
- This has one solution at \(x = 0\).

- For \(x \in [0, \pi]\):
- Here, \(y = |\sin x| = \sin x\).
- The equation becomes \(\sin x = x\).
- This has one solution at \(x = 0\) and another solution in \((0, \pi)\).

2. **Total solutions**:
- From the analysis, we find:
- One solution at \(x = 0\) from both intervals.
- One additional solution in \((0, \pi)\).
- The symmetry of the sine function gives us another solution in \((-π, 0)\).

### Conclusion

Thus, the total number of ordered pairs \((x, y)\) satisfying the equations is **3**.