1. Visualizing the Inverse Sine

[ Execute the code resource section. Although there will be no output immediately, these definitions are used below.]

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restart: with(plots): arcsinplot := proc( asin)    local A,B,C,E,F,G, H,J, theta, c1, c2, c3;    theta := arcsin(asin);    c1 := COLOR(RGB, 1, .75, .2 );    c2 := COLOR(RGB, .7, .4, .1);    c3 := COLOR(RGB, 1, .2, .1);    A  := plot(sin(x), x = -Pi..Pi,        color = c3, tickmarks = [1,1]);    B  := plot(sin(x), x = -Pi/2..Pi/2,    color = c2, thickness = 4);    C  := polygonplot( [[-Pi/2,-1],[Pi/2,-1],[Pi/2,1],[-Pi/2,1],[-Pi/2,-1]],                 color = c1, style = patchnogrid );    E  := plot( [[-Pi/2,-1],[Pi/2,-1],[Pi/2,1],[-Pi/2,1],[-Pi/2,-1]],                  color = orange, linestyle = 3);    F  := plottools[arrow]( [0,asin], [theta, asin], .05, .10, .15,          color = red);    G  := plottools[arrow]( [theta,asin], [theta, 0],.10,.20,.10,          color = red);    H := textplot([theta, -.15,  evalf(theta,3)],align={ABOVE, CENTER});    J := textplot([  -.2, asin,  evalf(asin, 3)],align={ABOVE, LEFT}); display([F,G,A,B,C,E,H,J ]); end proc:

Warning, the name changecoords has been redefined

 
           Example  1.1 : Compute arcsin( 0.5 )

>	arcsinplot( .5); arcsin( .5 );

            Example 1.2 : Compute arccsin( 0.9 )

>	arcsinplot(.9); arcsin( .9);

            Example 1.3 : Simplify arcsin( -0.3).

>	arcsinplot(-.8); arcsin(-.8);

 2. Visualizing the Inverse Cosine

[ Execute the code resource section. Although there will be no output immediately, these definitions are used below.]

>

restart: with(plots): arccosplot := proc( acos )    local A,B,C,E,F,G, H,J, theta, c1, c2, c3;    theta := arccos(acos);    c1 := COLOR(RGB, .6, .6, .85 );    c2 := COLOR(RGB, .3, .4, .87);    c3 := COLOR(RGB, .1, .2, .6);    A  := plot(cos(x), x = -Pi/2..3*Pi/2, color = c3, tickmarks = [0,0]);    B  := plot(cos(x), x =  0..Pi,    color = c2, thickness = 4);    C  := polygonplot( [[0,-1],[Pi,-1],[Pi,1],[0,1],[0,-1]],                 color = c1, style = patchnogrid );    E  := plot( [[0,-1],[Pi,-1],[Pi,1],[0,1],[0,-1]],                  color = navy, linestyle = 3);    F  := plottools[arrow]( [0,acos], [theta, acos],  .05, .10, .15,          color = navy);    G  := plottools[arrow]( [theta,acos], [theta, 0], .10, .20, .18,          color = navy);    H := textplot([theta, -.15,  evalf(theta,3)], align={ABOVE, CENTER});    J := textplot([  -.2, acos,  evalf( acos, 3)],align={ABOVE, LEFT}); display([A,B,E,F,G,C,H,J ]); end proc:

Warning, the name changecoords has been redefined

            Example 2.1  : Compute arccos(0.3).

>	arccosplot(.3); arccos(.3);

Noticethe difference between what happens when we compute arcsine of a negative number and the arccosine of a negative.

            Example 2.2 : Compute arccos( -0.6).

>	arccosplot(-.6); arccos(-.6);

            Example 2.3 : Compute arccos(0.8).

>	arccosplot(.8); arccos(.8);

 3. Visualizing the Inverse Tangent

[ Execute the code resource section. Although there will be no output immediately, these definitions are used below.]

  

>

restart: with(plots): arctanplot := proc( a)    local A,B,C,E,F,G, H,J, theta, c1, c2, c3, L,R,m;    theta := arctan(a);       L:= -Pi/2 + .01;    R:= Pi/2 - .01; m := 12;    c1 := COLOR(RGB, .75,  95, .75 );    c2 := COLOR(RGB, .4, .9, .4);    c3 := COLOR(RGB, .2, .5, .2);    A  := plot(tan(x), x = -Pi..Pi, y = -m..m,  color = c3,               tickmarks = [1,1], discont = true);    B  := plot(tan(x), x = L..R, y = -m..m,  color = c2, thickness = 4);    C  := polygonplot( [[L,-m],[R,-m],[R,m],[L,m],[L,-m]],                 color = c1, style = patchnogrid );    E  := plot( [[L,-m],[R,-m],[R,m],[L,m],[L,-m]],                  color = green, linestyle = 3);    F  := plottools[arrow]( [0,a], [theta, a], .4, .6, .15,          color = c3);    G  := plottools[arrow]( [theta,a], [theta, 0],.08,.15,.15,          color = c3);    H := textplot([theta, -.8,  evalf(theta,3)],align={ABOVE, CENTER});    J := textplot([  -.2,   a,   evalf(a, 3)],align={ABOVE, LEFT}); display([F,G,A,B,C,E,H,J ]); end proc:

Warning, the name changecoords has been redefined

 


           Example 3.1   :  Compute arctan(4.5).

>	arctanplot(4.5); arctan(4.5);

            Example 3.2  : Compute arctan( 10).

>	arctanplot(10); arctan( 10);

            Example 3.3  : Simplify arctan(-6).

>	arctanplot(-6); arctan(-6);