Trigonometric functions example) and another of the sides remains the same. If the hypotenuse is twice as long, so are the sides. It is these ratios that the trigonometric functions express. To define the trigonometric functions for the angle "A", start with any right triangle that contains the angle "A". The three sides of the triangle are named as follows: In ordinary Euclidean geometry, according to the triangle postulate, the inside angles of every triangle total 180° ( radians). Therefore, in a right-angled triangle, the two non-right angles total 90° ( radians), so each of these angles must be in the range
Triangle center Since rotations and translations may be regarded as double reflections they too must preserve triangle centers. These invariance properties provide justification for the definition. Some other names for dilation are uniform scaling, isotropic scaling, homothety, and homothecy. The study of triangle centers traditionally is concerned with Euclidean geometry, but triangle centers can also be studied in hyperbolic geometry. Using gyrotrigonometry, expressions for trigonometric barycentric coordinates can be calculated that have the same form for both Euclidean and hyperbolic geometry. In order for the expressions to coincide, the expressions must "not" encapsulate the specification of the angle sum being 180 degrees.