Final answer:
The value of -sin⁻¹(√2/2) in the fourth quadrant corresponds to the angle -315° or -7π/4 radians.
Step-by-step explanation:
To find the value of -sin⁻¹(√2/2) in the fourth quadrant, we need to understand the properties of the sine inverse function and the characteristics of the trigonometric functions in different quadrants.
The inverse sine function, also known as arcsin, returns the angle whose sine is the given number. However, the range of the arcsin function is typically from -π/2 to π/2 (or -90° to 90°), covering the first and fourth quadrants for positive angles, and the first and second quadrants for negative angles.
Since sine is positive in the first and second quadrants and negative in the third and fourth quadrants, the expression '-sin⁻¹(√2/2)' implies that we are looking for an angle with a negative sine value (√2/2 is positive, hence the preceding minus sign for adjustments), which falls within the fourth quadrant. The principal value (positive angle) for sin⁻¹(√2/2) is 45° or π/4 radians, because sin(45°) = √2/2. Hence, the fourth quadrant angle corresponding to this would be 360° - 45° which is 315° or 7π/4 radians. However, since the negative sign is also applied, we negate this angle resulting in -315° or -7π/4 radians as the final answer.