If the maximum value of x which satisfies the inequality `sin^(-1)(sinx) ge cos^(-1)(sinx)" for "x in (pi)/(2), 2pi" is " lambda,` then `(2lambda)/(3)` is equal to (take `pi=3.14`)

Answer

To solve the inequality \( \sin^{-1}(\sin x) \geq \cos^{-1}(\sin x) \) for \( x \) in the interval \( \left(\frac{\pi}{2}, 2\pi\right) \), we will follow these steps:

### Step 1: Understand the relationship between \( \sin^{-1} \) and \( \cos^{-1} \)
We know that:
\[ \sin^{-1}(y) + \cos^{-1}(y) = \frac{\pi}{2} \]
for \( y \) in the range \([-1, 1]\). Thus, we can rewrite the inequality:
\[ \sin^{-1}(\sin x) \geq \cos^{-1}(\sin x) \implies \sin^{-1}(\sin x) \geq \frac{\pi}{2} - \sin^{-1}(\sin x) \]

### Step 2: Rearranging the inequality
Rearranging gives:
\[ 2\sin^{-1}(\sin x) \geq \frac{\pi}{2} \]
Dividing both sides by 2:
\[ \sin^{-1}(\sin x) \geq \frac{\pi}{4} \]

### Step 3: Finding the range of \( \sin x \)
The function \( \sin^{-1}(\sin x) \) is equal to \( x \) for \( x \) in the range \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \) and \( \pi - x \) for \( x \) in the range \( \left(\frac{\pi}{2}, \frac{3\pi}{2}\right) \). Therefore, we need to analyze the intervals:

1. For \( x \in \left(\frac{\pi}{2}, \frac{3\pi}{2}\right) \):
\[ \sin^{-1}(\sin x) = \pi - x \]

2. For \( x \in \left(\frac{3\pi}{2}, 2\pi\right) \):
\[ \sin^{-1}(\sin x) = x - 2\pi \]

### Step 4: Solving the inequality
Now we solve the inequality \( \sin^{-1}(\sin x) \geq \frac{\pi}{4} \).

1. For \( x \in \left(\frac{\pi}{2}, \frac{3\pi}{2}\right) \):
\[ \pi - x \geq \frac{\pi}{4} \implies x \leq \pi - \frac{\pi}{4} = \frac{3\pi}{4} \]

2. For \( x \in \left(\frac{3\pi}{2}, 2\pi\right) \):
\[ x - 2\pi \geq \frac{\pi}{4} \implies x \geq 2\pi + \frac{\pi}{4} \quad (\text{not valid in this interval}) \]

### Step 5: Finding the maximum value of \( x \)
The maximum value of \( x \) satisfying the inequality in the interval \( \left(\frac{\pi}{2}, 2\pi\right) \) is \( \frac{3\pi}{4} \).

### Step 6: Assigning the value to \( \lambda \)
Thus, we have:
\[ \lambda = \frac{3\pi}{4} \]

### Step 7: Calculate \( \frac{2\lambda}{3} \)
Now we need to calculate:
\[ \frac{2\lambda}{3} = \frac{2 \cdot \frac{3\pi}{4}}{3} = \frac{2\cdot3\pi}{4\cdot3} = \frac{2\pi}{4} = \frac{\pi}{2} \]

### Step 8: Substitute \( \pi = 3.14 \)
Now substituting \( \pi \):
\[ \frac{\pi}{2} = \frac{3.14}{2} = 1.57 \]

Thus, the final answer is:
\[ \frac{2\lambda}{3} = 1.57 \]