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Showing content from https://dlmf.nist.gov/4.15 below:

§4.15 Graphics ‣ Trigonometric Functions ‣ Chapter 4 Elementary 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 Figure 4.15.1: sin ⁡ x and cos ⁡ x . Magnify Figure 4.15.2: Arcsin ⁡ x and Arccos ⁡ x . Principal values are shown with thickened lines. Magnify Figure 4.15.3: tan ⁡ x and cot ⁡ x . Magnify Figure 4.15.4: arctan ⁡ x and arccot ⁡ x . Only principal values are shown. arccot ⁡ x is discontinuous at x = 0 . Magnify Figure 4.15.5: csc ⁡ x and sec ⁡ x . Magnify Figure 4.15.6: arccsc ⁡ x and arcsec ⁡ x . 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 − 1 2 ⁢ π < ℜ ⁡ z < 1 2 ⁢ π onto the whole w -plane cut along the real axis from − ∞ to − 1 and 1 to ∞ , where w = sin ⁡ z and z = arcsin ⁡ w (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 , ∞ ) .

      (i) z -plane                                                  (ii) w -plane Figure 4.15.7: Conformal mapping of sine and inverse sine. w = sin ⁡ z , z = arcsin ⁡ w . 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.

Figure 4.15.9: arcsin ⁡ ( x + i ⁢ y ) (principal value). There are branch cuts along the real axis from − ∞ to − 1 and 1 to ∞ . Magnify 3D Help Figure 4.15.11: arctan ⁡ ( x + i ⁢ y ) (principal value). There are branch cuts along the imaginary axis from − i ⁢ ∞ to − i and i to i ⁢ ∞ . Magnify 3D Help Figure 4.15.13: arccsc ⁡ ( x + i ⁢ y ) (principal value). There is a branch cut along the real axis from − 1 to 1 . Magnify 3D Help

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

4.15.1 cos ⁡ ( x + i ⁢ y ) = sin ⁡ ( x + 1 2 ⁢ π + i ⁢ y ) , 4.15.2 cot ⁡ ( x + i ⁢ y ) = − tan ⁡ ( x + 1 2 ⁢ π + i ⁢ y ) , 4.15.3 sec ⁡ ( x + i ⁢ y ) = csc ⁡ ( x + 1 2 ⁢ π + i ⁢ y ) ,

they can be obtained by translating the surfaces shown in Figures 4.15.8, 4.15.10, 4.15.12 by − 1 2 ⁢ π parallel to the x -axis, and adjusting the phase coloring in the case of Figure 4.15.10.

The corresponding surfaces for arccos ⁡ ( x + i ⁢ y ) , arccot ⁡ ( x + i ⁢ y ) , arcsec ⁡ ( x + i ⁢ y ) 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).

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