3.9 HW Problems

Table of Contents

Khan Academy Videos: Differentiating Inverse Trig Functions

<aside> ☝🏾 You can determine if a function is even or odd if…

<aside> 🗒️ an even function has reflection symmetry about the y-axis

</aside>

<aside> 🗒️ an odd function has rotational symmetry about the origin

</aside>

</aside>

Inverses of tan x, cot x, sec x, and csc x

• The graphs of these 4 basic inverse trigonometric functions are shown below.

We got these graphs by reflecting the graphs of the restricted trig functions through the line $y=x$

• Let’s take a closer look at the arctangent, arcotangent, arcsecant, and arccosecant functions:
  • the arctangent of x is a radian angle whose tangent is x
  • the arcotangent of x is an angle whose cotangent is x
  • the arcsecant of x is an angle whose arcsecant is x
  • the arccosecant of x is an angle whose arcsecant is x
  • …repetitive but you get the point :)

Definitions

(we use open or half-open intervals to avoid values for when the functions are undefined)

• $y=\tan^{-1}x$ is the number in $(-\pi/2, \pi/2)$ for which $\tan y=x$
• $y=\cot^{-1}x$ is the number in $(0,\pi)$ for which $\cot y=x$
• $y=\sec^{-1}x$ is the number in $[0,\pi/2)\cup(\pi/2,\pi]$ for which $\sec y=x$
• $y=\csc^{-1}x$ is the number in $[-\pi/2,0)\cup(0,\pi/2]$ for which $\csc y=x$

Example

• The figures show two values of $\tan^{-1}x$