Ultimate fate of the universe eventually stops the expansion of the universe, after which it starts to contract until all matter in the universe collapses to a point, a final singularity termed the "Big Crunch", the opposite of the Big Bang. Some new modern theories assume the universe may have a significant amount of dark energy, whose repulsive force may be sufficient to cause the expansion of the universe to continue forever—even if formula_5. If formula_7, the geometry of space is open, i.e., negatively curved like the surface of a saddle. The angles of a triangle sum to less than 180 degrees, and lines that
Proofs of trigonometric identities division identity They remain valid for angles greater than 90° and for negative angles. Or Two angles whose sum is π/2 radians (90 degrees) are "complementary". In the diagram, the angles at vertices A and B are complementary, so we can exchange a and b, and change θ to π/2 − θ, obtaining: Identity 1: The following two results follow from this and the ratio identities. To obtain the first, divide both sides of formula_21 by formula_23; for the second, divide by formula_24. Similarly Identity 2: The following accounts for all three reciprocal functions. Proof 2: Refer to the triangle