Inverse Trigonometric Functions Study Notes

Definition: If $f$ is a one-to-one function with domain $A$ and range $B$ , then its inverse $f^{-1}$ is the function with domain $B$ and range $A$ defined by:
$f^{-1}(x) ext{ such that } f(y) = x$

One-to-One Requirement:
- A function must be one-to-one to have an inverse.
- The trigonometric functions are not one-to-one in their standard forms, hence they do not have inverses.
- We can restrict the domains of trigonometric functions to create one-to-one functions.

Definition:
- The inverse sine function is denoted as $y = ext{sin}^{-1}(x)$ which implies:
 $y = ext{sin}^{-1}(x) ext{ if and only if } x = ext{sin}(y) ext{ where } -1 ightarrow x ightarrow 1$
Domain:
- $D = [-1, 1]$
Range:
- $R = \bigg[ - rac{ u}{2}, rac{ u}{2} \bigg]$
Graph:
- The graph of $y = ext{sin}(x)$ shows that inverses can be found within the specified range.

Definition:
- The inverse cosine function is denoted as $y = ext{cos}^{-1}(x)$ if and only if:
  $y = ext{cos}^{-1}(x) ext{ if and only if } x = ext{cos}(y)$
Domain of x:
- $[-1, 1]$
Range of y:
- $[0, u]$
Graph:
- The representation of the inverse cosine function can be displayed similarly to the sine function.

Definition:
- The inverse tangent function is represented by:
  $y = ext{tan}^{-1}(x) ext{ if and only if } x = ext{tan}(y)$
Domain:
- $(- u, u)$
Range:
- $(- rac{ u}{2}, rac{ u}{2})$
Graph:
- The graph shows periodic tendencies similar to that of tangent but reflects inversely.

Definition:
- The inverse cotangent function is given by:
  $y = ext{cot}^{-1}(x) ext{ if and only if } x = ext{cot}(y)$
Domain:
- $(- u, u)$
Range:
- $(0, u)$
Graph:
- Similar periodic graph to tangent but inversely plotted.

Definition:
- The inverse secant function can be defined as:
  $y = ext{sec}^{-1}(x) ext{ if and only if } x = ext{sec}(y)$
Domain:
- $(- u, -1] \bigcup [1, u)$
Range:
- $[0, u]$
Graph:
- Graphically portrayed with restrictions at the boundaries of the domain.

Definition:
- The inverse cosecant function is described as:
  $y = ext{csc}^{-1}(x) ext{ if and only if } x = ext{csc}(y)$
Domain:
- $(- u, -1] \bigcup [1, u)$
Range:
- $[- rac{ u}{2}, 0) \bigcup (0, rac{ u}{2}]$
Graph:
- Exhibits traits similar to secant with domain restrictions reflected in the graph.

Inverse Functions:
- Complete table detailing domains and ranges:
- $y = ext{sin}^{-1}(x)$ : Domain $[-1,1]$ , Range $[- rac{ u}{2}, rac{ u}{2}]$
- $y = ext{cos}^{-1}(x)$ : Domain $[-1,1]$ , Range $[0, u]$
- $y = ext{tan}^{-1}(x)$ : Domain $(- u, u)$ , Range $(- rac{ u}{2}, rac{ u}{2})$
- $y = ext{cot}^{-1}(x)$ : Domain $(- u, u)$ , Range $(0, u)$
- $y = ext{sec}^{-1}(x)$ : Domain $(- u, -1] \bigcup [1, u)$ , Range $[0, u]$
- $y = ext{csc}^{-1}(x)$ : Domain $(- u, -1] \bigcup [1, u)$ , Range $[- rac{ u}{2}, 0) \bigcup (0, rac{ u}{2}]$

Explanation:
- Evaluating these functions involves finding an angle whose trigonometric value equals x. This angle must exist in the range of the inverse function.
Example:
- Finding $sin^{-1}( rac{1}{2})$ involves knowing $y$ so that $ ext{sin}(y) = rac{1}{2}$ under accepted ranges.
Special Consideration:
- Remember: "the arcsine of $x$ is the angle whose sine is $x$ ."

These angles are referenced for quick evaluations of the standard inverse trigonometric values known from various geometric contexts.

Example of evaluation:
- To find $cos^{-1}( rac{1}{2})$ , angle is $60°$ .
- Hence, the value is $y = cos^{-1}( rac{1}{2}) = 60°$ .

Table of Values:

Angle (Degrees)	$sin$	$cos$	$tan$	$csc$	$sec$	$cot$
0	0	1	0	Undef	1	Undef
$30°$	$ rac{1}{2}$	$ rac{
u}{2}$	$ rac{
u}{3}$	2	$ rac{2}{
u}$	$<br></p></td><td colspan="1" rowspan="1"><p></p></td><td colspan="1" rowspan="1"><p></p></td><td colspan="1" rowspan="1"><p></p></td><td colspan="1" rowspan="1"><p></p></td><td colspan="1" rowspan="1"><p></p></td><td colspan="1" rowspan="1"><p></p></td></tr><tr><td colspan="1" rowspan="1"><p>u$
$45°$	$ rac{
u}{2}$	$ rac{
u}{2}$	1	$<br></p></td><td colspan="1" rowspan="1"><p></p></td><td colspan="1" rowspan="1"><p></p></td><td colspan="1" rowspan="1"><p></p></td><td colspan="1" rowspan="1"><p></p></td><td colspan="1" rowspan="1"><p></p></td></tr><tr><td colspan="1" rowspan="1"><p>u$	$<br></p></td><td colspan="1" rowspan="1"><p></p></td><td colspan="1" rowspan="1"><p></p></td><td colspan="1" rowspan="1"><p></p></td><td colspan="1" rowspan="1"><p></p></td><td colspan="1" rowspan="1"><p></p></td><td colspan="1" rowspan="1"><p></p></td></tr><tr><td colspan="1" rowspan="1"><p>u$	1
$60°$	$ rac{
u}{2}$	$ rac{1}{2}$	$ rac{2}{
u}$	$ rac{2}{
u}$	2	$ rac{
u}{3}$
$90°$	1	0	Undef	1	Undef	0

Inverse Functions must satisfy:
- $f(f^{-1}(x)) = x$
- Properties are contingent upon domain and range limitations.

$ext{tan}[ ext{arctan}(-5)] = -5$ (Value lies in domain.)
Response for out-of-range elements must include coterminal angles.
- Example: $ext{sin}^{-1}(- rac{3 ext{π}}{5})$ converts as $- rac{ ext{π}}{3}$ for values acceptable within inverse function dimensions.
If unable to define, specify: "not defined within specified domain".

Understanding inverse trigonometric functions offers vital insights into solving and simplifying problems across various mathematical contexts and applications, particularly in applied sciences and engineering.