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The concept of the domain of an inverse function is crucial when dealing with trigonometric functions like the inverse sine function, also known as arcsine, denoted by \(sin^{-1}(x)\). The domain refers to the set of all input values for which the function is defined. For the inverse sine function, this domain is the interval \[ -1, 1 \], meaning \(sin^{-1}(x)\) can only take on values within this range.

This is because the sine function, which gives the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle, can only have values between -1 and 1. Thus, when we consider the inverse of this function, we're looking for an angle whose sine gives a certain value \(x\), but this value must be realistic within the context of the sine function's output. Hence, we cannot have values outside of \[ -1, 1 \] for \(sin^{-1}(x)\). Any values outside this range would not correspond to any angle, making the function undefined.

The increasing intervals of a function are the regions where the output value of the function increases as the input value increases. In the case of the inverse sine function, it's particularly simple: it is an increasing function over its entire domain. This means that for every pair of numbers \(x_1\) and \(x_2\) within the domain \[ -1, 1 \], if \(x_1 < x_2\), then \(sin^{-1}(x_1) < sin^{-1}(x_2)\).

Why is this the case? The answer lies in the fundamental shape of the sine function, which arcs upwards in its first cycle from 0 to \( \pi \). When we take the inverse of this function, we are reflecting it across the line \(y = x\), but the increasing nature of this particular section of the sine curve remains intact in the inverse. Therefore, the inverse sine function does not have any intervals where it decreases; it consistently ascends between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \) as \(x\) moves from -1 to 1.

The Principal Value Branch for the inverse sine function describes the primary interval of angles for which the inverse sine's output is considered the most meaningful or 'principal'. For \(sin^{-1}(x)\), the principal value branch is the interval from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \), or \[ -\frac{\pi}{2}, \frac{\pi}{2} \] in interval notation. This is because the sine function is originally defined for angles, and the angle range specified here corresponds to the angles in a right triangle.

The choice of this interval isn't arbitrary. This range is selected because, within this interval, each value of \(x\) in the domain \[ -1, 1 \] corresponds to exactly one angle, ensuring the function is one-to-one and hence has an inverse. Outside this range, the sine function starts to repeat values, thus losing the one-to-one property. It is important to remember that the concept of a principal value is closely related to the function's invertibility — a function can only have an inverse if it is one-to-one and does not repeat output values for different inputs within a given interval.