`sin^(-1)(x^2/4+y^2/9)+cos^(-1)(x/(2sqrt2)+y/(3sqrt2)-2)`

Options

A. `(pi)/(2)`
B. `pi`
C. `(pi)/(sqrt(2))`
D. `(3pi)/(2)`

Answer

To solve the expression \( \sin^{-1}\left(\frac{x^2}{4} + \frac{y^2}{9}\right) + \cos^{-1}\left(\frac{x}{2\sqrt{2}} + \frac{y}{3\sqrt{2}} - 2\right) \), we can follow these steps:

### Step 1: Substitute Variables
Let \( x = 2 \cos \theta \) and \( y = 3 \sin \theta \). This substitution simplifies the trigonometric expressions.

### Step 2: Substitute in the Expression
Substituting the values of \( x \) and \( y \) into the expression gives:
\[ \sin^{-1}\left(\frac{(2 \cos \theta)^2}{4} + \frac{(3 \sin \theta)^2}{9}\right) + \cos^{-1}\left(\frac{2 \cos \theta}{2\sqrt{2}} + \frac{3 \sin \theta}{3\sqrt{2}} - 2\right) \]

### Step 3: Simplify the First Term
The first term simplifies as follows:
\[ \sin^{-1}\left(\frac{4 \cos^2 \theta}{4} + \frac{9 \sin^2 \theta}{9}\right) = \sin^{-1}(\cos^2 \theta + \sin^2 \theta) = \sin^{-1}(1) \]
Since \( \cos^2 \theta + \sin^2 \theta = 1 \), we have:
\[ \sin^{-1}(1) = \frac{\pi}{2} \]

### Step 4: Simplify the Second Term
Now, simplify the second term:
\[ \cos^{-1}\left(\frac{2 \cos \theta}{2\sqrt{2}} + \frac{3 \sin \theta}{3\sqrt{2}} - 2\right) = \cos^{-1}\left(\frac{\cos \theta}{\sqrt{2}} + \frac{\sin \theta}{\sqrt{2}} - 2\right) \]
This can be rewritten as:
\[ \cos^{-1}\left(\frac{\cos \theta + \sin \theta}{\sqrt{2}} - 2\right) \]

### Step 5: Analyze the Range
The expression \( \frac{\cos \theta + \sin \theta}{\sqrt{2}} \) can take values from \(-\sqrt{2}\) to \(\sqrt{2}\). Therefore, the term \( \frac{\cos \theta + \sin \theta}{\sqrt{2}} - 2 \) will range from \(-\sqrt{2} - 2\) to \(\sqrt{2} - 2\).

### Step 6: Determine Validity of the Argument
The argument of \( \cos^{-1} \) must lie between -1 and 1. Thus, we need to check:
\[ -1 \leq \frac{\cos \theta + \sin \theta}{\sqrt{2}} - 2 \leq 1 \]
This gives us:
\[ 1 \leq \frac{\cos \theta + \sin \theta}{\sqrt{2}} \leq 3 \]
Since \( \cos \theta + \sin \theta \) can only achieve a maximum of \(\sqrt{2}\), the only feasible solution is when:
\[ \frac{\cos \theta + \sin \theta}{\sqrt{2}} = 1 \]
This implies:
\[ \cos \theta + \sin \theta = \sqrt{2} \]

### Step 7: Find \( \theta \)
The equation \( \cos \theta + \sin \theta = \sqrt{2} \) leads to:
\[ \theta = \frac{\pi}{4} \]

### Step 8: Substitute Back
Substituting \( \theta = \frac{\pi}{4} \) back into the expression gives:
\[ \frac{\pi}{2} + \cos^{-1}(0 - 2) \]
This simplifies to:
\[ \frac{\pi}{2} + \cos^{-1}(-1) = \frac{\pi}{2} + \pi = \frac{3\pi}{2} \]

### Final Answer
Thus, the value of the given expression is:
\[ \frac{3\pi}{2} \]