About the Project
4 Elementary FunctionsTrigonometric Functions

§4.15 Graphics

Contents
  1. §4.15(i) Real Arguments
  2. §4.15(ii) Complex Arguments: Conformal Maps
  3. §4.15(iii) Complex Arguments: Surfaces

§4.15(i) Real Arguments

See accompanying text
Figure 4.15.1: sinx and cosx. Magnify
See accompanying text
Figure 4.15.2: Arcsinx and Arccosx. Principal values are shown with thickened lines. Magnify
See accompanying text
Figure 4.15.3: tanx and cotx. Magnify
See accompanying text
Figure 4.15.4: arctanx and arccotx. Only principal values are shown. arccotx is discontinuous at x=0. Magnify
See accompanying text
Figure 4.15.5: cscx and secx. Magnify
See accompanying text
Figure 4.15.6: arccscx and arcsecx. Only principal values are shown. (Both functions are complex when 1<x<1.) Magnify

§4.15(ii) Complex Arguments: Conformal Maps

Figure 4.15.7 illustrates the conformal mapping of the strip 12π<z<12π onto the whole w-plane cut along the real axis from to 1 and 1 to , where w=sinz and z=arcsinw (principal value). Corresponding points share the same letters, with bars signifying complex conjugates. Lines parallel to the real axis in the z-plane map onto ellipses in the w-plane with foci at w=±1, and lines parallel to the imaginary axis in the z-plane map onto rectangular hyperbolas confocal with the ellipses. In the labeling of corresponding points r is a real parameter that can lie anywhere in the interval (0,).

See accompanying text
      (i) z-plane                                                  (ii) w-plane
A B C C¯ D D¯ E E¯ F
z 0 12π 12π+ir 12πir ir ir 12π+ir 12πir 12π
w 0 1 coshr+i0 coshri0 isinhr isinhr coshr+i0 coshri0 1
Figure 4.15.7: Conformal mapping of sine and inverse sine. w=sinz, z=arcsinw. Magnify

§4.15(iii) Complex Arguments: Surfaces

In the graphics shown in this subsection height corresponds to the absolute value of the function and color to the phase. See also About Color Map.

See accompanying text
Figure 4.15.8: sin(x+iy). Magnify 3D Help
See accompanying text
Figure 4.15.9: arcsin(x+iy) (principal value). There are branch cuts along the real axis from to 1 and 1 to . Magnify 3D Help
See accompanying text
Figure 4.15.10: tan(x+iy). Magnify 3D Help
See accompanying text
Figure 4.15.11: arctan(x+iy) (principal value). There are branch cuts along the imaginary axis from i to i and i to i. Magnify 3D Help
See accompanying text
Figure 4.15.12: csc(x+iy). Magnify 3D Help
See accompanying text
Figure 4.15.13: arccsc(x+iy) (principal value). There is a branch cut along the real axis from 1 to 1. Magnify 3D Help

The corresponding surfaces for cos(x+iy), cot(x+iy), and sec(x+iy) are similar. In consequence of the identities

4.15.1 cos(x+iy) =sin(x+12π+iy),
4.15.2 cot(x+iy) =tan(x+12π+iy),
4.15.3 sec(x+iy) =csc(x+12π+iy),

they can be obtained by translating the surfaces shown in Figures 4.15.8, 4.15.10, 4.15.12 by 12π parallel to the x-axis, and adjusting the phase coloring in the case of Figure 4.15.10.

The corresponding surfaces for arccos(x+iy), arccot(x+iy), arcsec(x+iy) can be visualized from Figures 4.15.9, 4.15.11, 4.15.13 with the aid of equations (4.23.16)–(4.23.18).