Graphing Inverse Trigonometric Functions

Key Questions

  • Throughout the following answer, I will assume that you are asking about trigonometry restricted to real numbers.

    Using Domain of arc sin xarcsinx

    Find arc sin (3)arcsin(3).

    33 is not in the domain of arc sinarcsin, (33 is not in the range of sinsin) so arc sin (3)arcsin(3) does not exist.

    Using Range of arc sin xarcsinx

    Find arc sin (1/2)arcsin(12).

    Although there are infinitely many tt with sin t = 1/2sint=12, the range of arc sinarcsin restricts the value to those tt with (-pi)/2 <= t <= pi/2π2tπ2, So the value we want is pi/6π6.

    Using the quadrant
    This is the same as using the range, but it involves thinking about the problem more geometrically.

    Find arc sin (1/2)arcsin(12).

    arc sin (1/2)arcsin(12) is a number (an angle) in quadrant I or IV. It is the tt with smallest absolute value (the shortest path from the initial side).

    Again, arc sin (1/2) = pi/6arcsin(12)=π6.

  • For a trig function, the range is called "Period"

    For example, the function f(x) = cos xf(x)=cosx has a period of 2pi2π; the function f(x) = tan xf(x)=tanx has a period of piπ. Solving or graphing a trig function must cover a whole period.

    The range depends on each specific trig function.
    For example, the inverse function f(x) = 1/(cos x) = sec xf(x)=1cosx=secx has as period 2pi2π.

    Its range varies from (+infinity) to Minimum 11 then back to (+infinity), between (-pi/2π2 and pi/2π2).

    Its range also varies from (-infinity) to Max -1 then back to to (-infinity), between (pi/2π2 and 3pi/23π2).

  • Since the graphs of f(x)f(x) and f'(x) are symmetric about the line y=x, start with the graph of a trigonometric function with an appropriate restricted domain, then reflect it about the line y=x.

    (Caution: Their domains must be restricted to an appropriate interval so that their inverses exist.)


    Let us sketch the graph of y=sin^{-1}x.

    The graph of y=sinx on [-pi/2, pi/2] looks like:

    enter image source here

    By reflecting the graph above about the line y=x,

    enter image source here

    The curve in purple is the graph of y=sin^{-1}x.

    The graphs of other inverse trigonometric functions can be obtained similarly.


    I hope that this was helpful.

Questions