L> The Inverse Sine Function Understanding theInverse Sine Function"> The Function y = sin-1x = arcsin x and its Graph: Because y = sin -1x is the inverse of the feature y = sin x, the attribute y = sin-1x if and also just if sin y = x. But, considering that y = sin x is not one-to-one, its domain need to be minimal in order that y = sin-1x is a role. To acquire the graph of y = sin-1x, start through a graph of y = sin x. Restrict the doprimary of the feature to a one-to-one area - typically" width="57" height="39" align="absmiddle"> is supplied (highlighted in red at right) for sin-1x. This leaves the range of the limited attribute unadjusted as <-1, 1>. Reflect this graph throughout the line y = x to get the graph of y = sin-1x (y = arcsin x), the black curve at right. Notice that y = sin-1x has domain <-1, 1> and also variety " width="56" height="39" align="absmiddle">. It is strictly enhancing on its entire doprimary. So, when you ask your calculator to graph y = sin-1x, you get the graph shown at ideal. (The viewing window is <-2, 2> x <-2, 2>.) ## Assessing y = sin-1x:

### Example 1: Evaluate sin-1(1/2)

Most people are more acquainted (and also even more comfortable) via the trigonometric functions than their inverses. Thus, the first step in evaluating this expression is to say that if y = sin-1(1/2), then sin y = 1/2. This basic trigonometric feature has actually an boundless variety of solutions: Five of these options are indicated by vertical lines on the graph of y = sin x below. So, is the worth of sin-1 (1/2) offered by the expressions above? No! It is vitally crucial to store in mind that the inverse sine function is a single-valued, one-to-one feature. Only one of the infinite number of remedies offered above is the outcome we desire. Which one? Remember that the range of sin-1x is " width="56" height="39" align="absmiddle">, which is shown in blue in the number above. It is really necessary to recognize the domain and range of the inverse trigonometric functions! (Why is this blue interval noted on the x-axis if it represents the selection of sin-1x? Due to the fact that the range of the inverse attribute equates to the domain of the principal attribute.) The just solution of y = sin x that drops within the compelled array is (the solid red line in the figure above). Thus, ### Example 2: What is   A unit-circle diagram is shown at appropriate. Notice that candidates for the solution include: However before, only one of these values is in the variety of sin-1x (" width="56" height="39" align="absmiddle">), so: You are watching: Y=sin^-1(x)

The Derivative of y = sin-1x: The derivative of y = sin-1 x is: (Click below for a derivation.) The graphs of y = sin-1 x and also its derivative is shown at best. The domain of y" is (-1. 1). Due to the fact that y = sin-1 x is constantly boosting, y" > 0 for all x in its doprimary.

## Integrals Involving the Inverse Sine Function Because , . This indicates that the arcsine function arises in discussions entailing integrals (and areas) of "reasonably normal looking" algebraic features.

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For instance: This is the shaded area displayed in the TI-89 display swarm at best. (The window is <-0.5, 1.1> x <0, 3>.)