The general solution of sin x = -1 is nπ + (-1)^n (3π/2), n ∈ Z

The principal solution of sin x = 0 lies in [-π/2, π/2]

• Both A and R are true and R is correct explanation of A
• Both A and R are true and R is not the correct explanation of A
• A is true but R is false
• D is false but R is true

The equation sin x = -1 has a general solution expressed as nπ + (-1)^n (3π/2), where n is any integer (n ∈ Z). This means that the sine function equals -1 at specific intervals along the x-axis. The principal solution for sin x = 0 is found within the interval [-π/2, π/2], which is x = 0.

Analyzing the Statements

• A: The general solution of sin x = -1 is nπ + (-1)^n (3π/2).
• R: The principal solution of sin x = 0 lies in [-π/2, π/2].

Evaluating the Truth of A and R

Both statements A and R are indeed true. However, R does not serve as a correct explanation for A, as they address different aspects of the sine function. A focuses on where sin x equals -1, while R pertains to where sin x equals 0.

Conclusion on the Options

Based on the analysis:

• Both A and R are true, but R does not explain A.

Thus, the correct choice is: "Both A and R are true and R is not the correct explanation of A."