Question

Find the exact value of each expression, if it is defined. Express your answer in radians. (If an answer is undefined, enter undefined.) (a) sin⁻¹(-2/2)

Answer 1

The expression
`sin⁻¹(-2/2)` is undefined.

Step-by-step explanation:

The expression
`sin⁻¹(-2/2)` involves finding the arcsine of -2/2 which simplifies to
`sin⁻¹(-1).` The arcsine function returns an angle whose sine is equal to the given value. However, the sine function's range is
`[-1, 1]`and there is no angle whose sine is -1 within this range. Therefore
`sin⁻¹(-2/2)` is undefined.

In mathematical terms sin
`⁻¹(-2/2`) implies finding an angle θ such that sin(θ) =
`-1/1`. The sine function is the ratio of the opposite side to the hypotenuse in a right-angled triangle. In this case, having a negative value for the sine implies a negative y-coordinate. However in the unit circle where the sine is defined, the y-coordinate is always between -1 and 1. Since there is no angle in the unit circle where the sine is -1 the expression sin
`⁻¹(-2/2`) is undefined.

In conclusion, the arcsine of
`-2/2`does not have a valid solution in the context of the unit circle or right-angled triangles. Therefore, the final answer is that sin
`⁻¹(-2/2)` is undefined, and there is no angle whose sine is -1 within the conventional range of the sine function.