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 x

    Find arc sin (3).

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

    Using Range of arc sin x

    Find arc sin (1/2).

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

    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).

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

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

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

    For example, the function f(x) = cos x has a period of 2pi; the function f(x) = tan x has a period of pi. 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 x has as period 2pi.

    Its range varies from (+infinity) to Minimum 1 then back to (+infinity), between (-pi/2 and pi/2).

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

  • Since the graphs of 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