Inverse Trigonometric Functions Study Notes

INVERSE TRIGONOMETRIC FUNCTIONS

  • 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

REQUIREMENTS FOR INVERSES

  • 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.

INVERSE SINE FUNCTION AND ITS GRAPH

  • 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:

    • $R = igg[ - rac{
      u}{2}, rac{
      u}{2} igg]$

  • Graph:

    • The graph of $y = ext{sin}(x)$ shows that inverses can be found within the specified range.

INVERSE COSINE FUNCTION AND ITS GRAPH

  • 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.

INVERSE TANGENT FUNCTION AND ITS GRAPH

  • 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.

INVERSE COTANGENT FUNCTION AND ITS GRAPH

  • 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.

INVERSE SECANT FUNCTION AND ITS GRAPH

  • Definition:

    • The inverse secant function can be defined as:
      y = ext{sec}^{-1}(x) ext{ if and only if } x = ext{sec}(y)

  • Domain:

    • $(-
      u, -1] igcup [1,
      u)$

  • Range:

    • $[0,
      u]$

  • Graph:

    • Graphically portrayed with restrictions at the boundaries of the domain.

INVERSE COSECANT FUNCTION AND ITS GRAPH

  • Definition:

    • The inverse cosecant function is described as:
      y = ext{csc}^{-1}(x) ext{ if and only if } x = ext{csc}(y)

  • Domain:

    • $(-
      u, -1] igcup [1,
      u)$

  • Range:

    • $[- rac{
      u}{2}, 0) igcup (0, rac{
      u}{2}]$

  • Graph:

    • Exhibits traits similar to secant with domain restrictions reflected in the graph.

DOMAIN AND RANGE OF INVERSE FUNCTIONS

  • 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 $(-
      u, -1] igcup [1,
      u)$, Range $[0,
      u]$

    • $y = ext{csc}^{-1}(x)$: Domain $(-
      u, -1] igcup [1,
      u)$, Range $[- rac{
      u}{2}, 0) igcup (0, rac{
      u}{2}]$

EVALUATING INVERSE TRIGRONOMETRIC FUNCTIONS

  • 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 $x$."

EVALUATING WITH SPECIAL AND QUADRANTAL ANGLES

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

EXAMPLES

  1. Example of evaluation:

    • To find $cos^{-1}( rac{1}{2})$, angle is $60°$.

    • Hence, the value is $y = cos^{-1}( rac{1}{2}) = 60°$.

TRIGONOMETRIC FUNCTIONS FOR SPECIAL ANGLES


  • 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}$

    $

    u$


    $45°$

    $ rac{

    u}{2}$

    $ rac{

    u}{2}$

    1

    $

    u$

    $

    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

    FUNCTION PROPERTY INVERSES

    • Inverse Functions must satisfy:

      • f(f^{-1}(x)) = x

      • Properties are contingent upon domain and range limitations.

    COMMON EXAMPLE SOLUTIONS

    1. ext{tan}[ ext{arctan}(-5)] = -5 (Value lies in domain.)

    2. 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.

    3. If unable to define, specify: "not defined within specified domain".

    CONCLUSION

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