Activity 4.3.3.
The goal of this activity is to understand key properties of the arcsine function in a way similar to our recent discussion of the arccosine function.
(a)
Using the definition of the arcsine function, what are the domain and range of the arcsine function?
(b)
Determine the following values exactly: \(\arcsin(-1)\text{,}\) \(\arcsin(-\frac{\sqrt{2}}{2})\text{,}\) \(\arcsin(0)\text{,}\) \(\arcsin(\frac{1}{2})\text{,}\) and \(\arcsin(\frac{\sqrt{3}}{2})\text{.}\)
(c)
On the axes provided, sketch a careful plot of the restricted sine function on the interval \([-\frac{\pi}{2},\frac{\pi}{2}]\) along with its corresponding inverse, the arcsine function. Label at least three points on each curve so that each point on the sine graph corresponds to a point on the arcsine graph. In addition, sketch the line \(y = t\) to demonstrate how the graphs are reflections of one another across this line.
(d)
True or false: \(\arcsin(\sin(5\pi)) = 5\pi\text{.}\) Write a complete sentence to explain your reasoning.
