4.3 Inverses of trigonometric functions

Introduction

The arccosine function

The arcsine function

Consider the plot of the standard cosine function in the following figure along with the emphasized portion of the graph on $[0,\pi]$.

Let $g$ be the function whose domain is $0 \le t \le \pi$ and whose outputs are determined by the rule $g(t) = \cos(t)$. Note well: $g$ is defined in terms of the cosine function, but because it has a different domain, it is not the cosine function.

What is the domain of $g$?

$\arccos(\frac{1}{2})$

$\arccos(\frac{\sqrt{2}}{2})$

$\arccos(\frac{\sqrt{3}}{2})$

$\arccos(-\frac{1}{2})$

$\arccos(-\frac{\sqrt{2}}{2})$

$\arccos(-\frac{\sqrt{3}}{2})$

$\arccos(-1)$

$\arccos(0)$

$\cos(\arccos(-\frac{1}{2}))$

$\arccos(\cos(\frac{7\pi}{6}))$

Using the definition of the arcsine function, what are the domain and range of the arcsine function?

Determine the following values exactly: $\arcsin(-1)$, $\arcsin(-\frac{\sqrt{2}}{2})$, $\arcsin(0)$, $\arcsin(\frac{1}{2})$, and $\arcsin(\frac{\sqrt{3}}{2})$.

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.

True or false: $\arcsin(\sin(5\pi)) = 5\pi$. Write a complete sentence to explain your reasoning.