Graph, Domain and Range of Arcsin function

The graph and the properties of the inverse trigonometric function arcsin are explored using an applet. You may want to look at an interactive tutorial on Inverse Trigonometric Functions before you work through the present tutorial.

The exploration is carried out by analyzing the effects of the parameters a, b, c and d included in the definition of arcsin as follows

f(x) = a arcsin(b x + c) + d

Interactive Tutorial

Change parameters a, b, c and d and click on the button 'draw' in the left panel below.

a =
-10+10

b =
-10+10

c =
-10+10

d =
-10+10



  1. Set the parameters to a = 1, b = 1, c = 0 and d = 0 to obtain

    f(x) = arcsin(x)


    Check that the domain of arcsin(x) is given by the interval [-1 , 1] and the range is given by the interval [-π/2 , +π/2](π/2 is approximately 1.57).

  2. Change parameter a and note how the graph of arcsin changes (vertical compression, stretching, reflection). How does it affect the range of the arcsin function?

    Does a change in parameter a changes the domain of arcsin?

  3. Change parameter b and note how the graph of arcsin changes (horizontal compression, stretching). Does a change in b affect the domain of arcsin? range?

  4. Change parameter c and note how the graph of arcsin changes (horizontal shift). Does a change in c affect the domain of arcsin? range?

  5. Change parameter d and note how the graph of arcsin changes (vertical shift). Does a change in d affect the range of arcsin? domain?

  6. If the range of arcsin(x) is given by the interval [-π/2 , π/2] what is the range of a*arcsin(x)? What is the range of a*arcsin(x)+ d?

  7. What is the domain and range of a*arcsin(bx + c)+ d?

Exercises

  1. Find the domain and range of f(x) = arcsin(x - 1) - 2 and graph f.

  2. Find the domain and range of g(x) = -arcsin(x + 1) + 2 and graph g.

  3. Find the domain and range of h(x) = -2arcsin(x + 1) -1 and graph h.

    More on Inverse Trigonometric Functions