Assertion (A) : `sin^(-1)(sin""(2pi)/(3))=(2pi)/(3)`
Reason (R ) : `sin^(-1)(sintheta)=theta`, if `thetain[(-pi)/(2),(pi)/(2)]`

Options

A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is NOT the correct explanation of A
C. A is true but R is false
D. A is false and R is True

Answer

To solve the assertion and reason provided in the question, we will analyze both the assertion (A) and the reason (R) step by step.

### Step 1: Understanding the Assertion (A)
The assertion states:
\[ \sin^{-1}\left(\sin\left(\frac{2\pi}{3}\right)\right) = \frac{2\pi}{3} \]
We need to check if this statement is true.

### Step 2: Determine the Value of \(\sin\left(\frac{2\pi}{3}\right)\)
The angle \(\frac{2\pi}{3}\) is in the second quadrant. The sine function is positive in the second quadrant. We can find:
\[ \sin\left(\frac{2\pi}{3}\right) = \sin\left(\pi - \frac{\pi}{3}\right) = \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \]

### Step 3: Substitute Back into the Assertion
Now we substitute this value back into the assertion:
\[ \sin^{-1}\left(\sin\left(\frac{2\pi}{3}\right)\right) = \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) \]

### Step 4: Evaluate \(\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)\)
The value of \(\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)\) is:
\[ \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3} \]
because \(\frac{\pi}{3}\) is in the range of \(\sin^{-1}\) which is \([- \frac{\pi}{2}, \frac{\pi}{2}]\).

### Step 5: Conclusion of Assertion
Thus, we have:
\[ \sin^{-1}\left(\sin\left(\frac{2\pi}{3}\right)\right) = \frac{\pi}{3} \neq \frac{2\pi}{3} \]
This means that the assertion (A) is **false**.

### Step 6: Understanding the Reason (R)
The reason states:
\[ \sin^{-1}(\sin \theta) = \theta, \text{ if } \theta \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \]
This statement is true, as it correctly describes the property of the inverse sine function.

### Final Conclusion
- Assertion (A) is **false**.
- Reason (R) is **true**.