8.3 Elliptic trigonometric functions and identities

Elliptic trigonometric functions are the non-Euclidean cousins of sine and cosine. They're defined on a curved plane, giving them unique properties like double periodicity and more complex graphs.

These functions play a key role in elliptic geometry, where triangles have angle sums greater than 180°. They're used in special versions of the law of cosines and sines for solving elliptic triangles.

Elliptic Trigonometric Functions

Definitions of elliptic trigonometric functions

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• Analogous to Euclidean trigonometric functions but defined on the elliptic plane which is a non-Euclidean geometry with constant positive curvature
• Elliptic sine function $\operatorname{sn}(u, k)$ defined as the ratio of the elliptic y-coordinate to the elliptic radius
  • $k$ represents the elliptic modulus which determines the shape of the elliptic functions (e.g., $k = 0.5$, $k = 0.8$)
• Elliptic cosine function $\operatorname{cn}(u, k)$ defined as the ratio of the elliptic x-coordinate to the elliptic radius
• Elliptic delta amplitude function $\operatorname{dn}(u, k)$ defined as the ratio of the elliptic radius to the semi-major axis of the ellipse

Fundamental identities in elliptic trigonometry

• Elliptic Pythagorean identity $\operatorname{sn}^2(u, k) + \operatorname{cn}^2(u, k) = 1$ relates the elliptic sine and cosine functions
• Elliptic angle addition formulas:
  1. $\operatorname{sn}(u + v, k) = \frac{\operatorname{sn}(u, k)\operatorname{cn}(v, k)\operatorname{dn}(v, k) + \operatorname{sn}(v, k)\operatorname{cn}(u, k)\operatorname{dn}(u, k)}{1 - k^2\operatorname{sn}^2(u, k)\operatorname{sn}^2(v, k)}$
  2. $\operatorname{cn}(u + v, k) = \frac{\operatorname{cn}(u, k)\operatorname{cn}(v, k) - \operatorname{sn}(u, k)\operatorname{sn}(v, k)\operatorname{dn}(u, k)\operatorname{dn}(v, k)}{1 - k^2\operatorname{sn}^2(u, k)\operatorname{sn}^2(v, k)}$
  3. $\operatorname{dn}(u + v, k) = \frac{\operatorname{dn}(u, k)\operatorname{dn}(v, k) - k^2\operatorname{sn}(u, k)\operatorname{sn}(v, k)\operatorname{cn}(u, k)\operatorname{cn}(v, k)}{1 - k^2\operatorname{sn}^2(u, k)\operatorname{sn}^2(v, k)}$

Comparing Elliptic and Euclidean Trigonometric Functions

Elliptic vs Euclidean trigonometric functions

• Periodicity differs: elliptic functions have two periods that depend on the elliptic modulus $k$, while Euclidean functions have one period
• Range similarities: elliptic sine and cosine functions have a range of [-1, 1], like Euclidean functions
  • Elliptic delta amplitude function has a range of [$\sqrt{1-k^2}$, 1]
• Graphs of elliptic functions are more complex and their shape depends on the value of the elliptic modulus $k$ (e.g., $k = 0.3$, $k = 0.7$)

Applications in elliptic geometry

• Elliptic triangles have the sum of their angles greater than 180 degrees, unlike Euclidean triangles
• Elliptic law of cosines relates the sides and angles of an elliptic triangle: $\cos a = \frac{\cos A + \cos B \cos C}{\sin B \sin C}$
  • $a$, $b$, and $c$ represent the sides and $A$, $B$, and $C$ are the corresponding opposite angles
• Elliptic law of sines relates the sides and angles of an elliptic triangle: $\frac{\sin A}{\sin a} = \frac{\sin B}{\sin b} = \frac{\sin C}{\sin c}$
• Solving elliptic triangles involves using the elliptic laws of cosines and sines, along with the elliptic trigonometric functions and identities, to find unknown angles or sides (e.g., given two sides and an angle, find the remaining side and angles)