Graphing Inverse Trigonometric Functions

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y=arcsinx graph - Related to C3 Mathematics Exams - Edexcel

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1 of 3 videos by Tiago Hands

Key Questions

  • 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:

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    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.

  • 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#).

  • It is best to start by graphing the parent inverse trig function. For example, if you graph the inverse tangent graph, you would have horizontal asymptotes where tangent has vertical asymptotes - which is where cosine is zero the first time in both directions - which is at - pi/2 and pi/2. You graph your horizontal asymptotes, plot the origin (because the tangent of zero is zero), then draw what looks pretty much like an #x^3# graphing laying on its side.

    Apply the transformations to the horizontal asymptotes and to the point at the origin, and draw the transformed graph the same way.

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