The complete set of values of x satisfying the inequality `sin^(-1)(sin 5) gt x^(2)-4x` is `(2-sqrt(lambda-2pi), 2+sqrt(lambda-2pi))`, then `lambda=`

Answer

To solve the inequality \( \sin^{-1}(\sin 5) > x^2 - 4x \), we will follow these steps:

### Step 1: Determine \( \sin^{-1}(\sin 5) \)

Since \( 5 \) is not in the range of \( \sin^{-1} \), we need to find the equivalent angle within the range of \( \sin^{-1} \). The range of \( \sin^{-1} \) is \( [-\frac{\pi}{2}, \frac{\pi}{2}] \).

The angle \( 5 \) radians lies between \( \frac{3\pi}{2} \) and \( \frac{5\pi}{2} \). To find the equivalent angle, we can subtract \( 2\pi \):

\[ \sin^{-1}(\sin 5) = 5 - 2\pi \]

### Step 2: Set up the inequality

Now we can rewrite the inequality:

\[ 5 - 2\pi > x^2 - 4x \]

Rearranging gives:

\[ x^2 - 4x + (2\pi - 5) < 0 \]

### Step 3: Solve the quadratic inequality

The quadratic can be expressed as:

\[ x^2 - 4x + (2\pi - 5) = 0 \]

Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):

Here, \( a = 1 \), \( b = -4 \), and \( c = 2\pi - 5 \).

Calculating the discriminant:

\[ D = (-4)^2 - 4 \cdot 1 \cdot (2\pi - 5) = 16 - 4(2\pi - 5) = 16 - 8\pi + 20 = 36 - 8\pi \]

Now we can find the roots:

\[ x = \frac{4 \pm \sqrt{36 - 8\pi}}{2} = 2 \pm \sqrt{9 - 2\pi} \]

### Step 4: Determine the intervals

The quadratic \( x^2 - 4x + (2\pi - 5) < 0 \) will be negative between its roots:

\[ x \in (2 - \sqrt{9 - 2\pi}, 2 + \sqrt{9 - 2\pi}) \]

### Step 5: Compare with the given solution

The problem states that the solution is of the form \( (2 - \sqrt{\lambda - 2\pi}, 2 + \sqrt{\lambda - 2\pi}) \).

By comparing:

\[ \sqrt{9 - 2\pi} = \sqrt{\lambda - 2\pi} \]

Squaring both sides gives:

\[ 9 - 2\pi = \lambda - 2\pi \]

Thus, we can solve for \( \lambda \):

\[ \lambda = 9 \]

### Final Answer

The value of \( \lambda \) is:

\[ \lambda = 9 \]

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