Find the Value of x + y + z
Given:
\[
\sin^{-1}x + \sin^{-1}y + \sin^{-1}z = \frac{3\pi}{2}
\]
Concept Used:
The principal value range of inverse sine is:
\[
\sin^{-1}t \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]
\]
Maximum value of each term = \( \frac{\pi}{2} \)
Maximum value of each term = \( \frac{\pi}{2} \)
Step 1:
Since the sum of three inverse sine functions is:
\[
\frac{3\pi}{2}
\]
this is the maximum possible sum.
Step 2:
This happens only when:
\[
\sin^{-1}x = \sin^{-1}y = \sin^{-1}z = \frac{\pi}{2}
\]
Step 3:
\[
\sin^{-1}x = \frac{\pi}{2} \Rightarrow x = 1
\]
\[
\sin^{-1}y = \frac{\pi}{2} \Rightarrow y = 1
\]
\[
\sin^{-1}z = \frac{\pi}{2} \Rightarrow z = 1
\]
Step 4: Required Value
\[
x + y + z = 1 + 1 + 1 = 3
\]
Final Answer:
\[
3
\]