Final answer:
1. Spherical line: A spherical line is a curve formed by the intersection of a sphere with a plane that passes through its center. Examples include the equator on Earth and lines of longitude. A non-example would be a straight line on a flat surface.
2. Law of Sines (spherical): The Law of Sines on a sphere states that for any spherical triangle, the ratio of the sines of the sides is equal to the ratio of the sines of their opposite angles. It's represented as sin(a) / sin(A) = sin(b) / sin(B) = sin(c) / sin(C). It doesn't hold in Euclidean geometry.
3. Hyperbolic sine function: The hyperbolic sine function, written as sinh(x), is a mathematical function that relates to hyperbolas. Examples include sinh(0) = 0 and sinh(1) ≈ 1.175. A non-example would be a trigonometric sine function.
Step-by-step explanation:
A spherical line refers to the curve formed by the intersection of a sphere and a plane. Visualize this as the equator on Earth or lines of longitude converging at the poles. A non-example would be a straight line on a flat surface like a paper or a computer screen, which isn't part of a spherical surface.
The Law of Sines applied to spherical geometry pertains to the relationship between the sides and angles in a spherical triangle. It states that the ratio of the sines of any two sides in a spherical triangle is equal to the ratio of the sines of their opposite angles. However, this law doesn't hold in Euclidean (planar) geometry, which sets it apart from the typical Law of Sines used in regular triangles.
The hyperbolic sine function, sinh(x), is a mathematical function related to hyperbolas, resembling the trigonometric sine function. For instance, sinh(0) = 0 and sinh(1) ≈ 1.175. It's different from the typical sine function, operating within the hyperbolic realm rather than the trigonometric. An example of this function is its behavior around certain values, while a non-example could be the regular sine function commonly used in trigonometry.