Gegenkathete

Gegenkathete is a term used in trigonometry to refer to the side of a right triangle that is opposite a given acute angle. It is one of the two legs (the non-hypotenuse sides) of the triangle. The other leg is called the Ankathete (the adjacent side), while the remaining side is the Hypotenuse, which is opposite the right angle.

For a given acute angle α in a right triangle, the Gegenkathete is the side that does not touch the angle α. The Ankathete is the leg that forms the angle together with the hypotenuse, and the Hypotenuse is the longest side.

In trigonometric terms, the Gegenkathete, Ankathete, and Hypotenuse determine the standard ratios:

- sin α = Gegenkathete / Hypotenuse

- cos α = Ankathete / Hypotenuse

- tan α = Gegenkathete / Ankathete

The Pythagorean theorem also relates the sides: Gegenkathete² + Ankathete² = Hypotenuse².

Notation can vary: in many conventions, the sides are labeled a and b as the legs and c as the hypotenuse, with a being the Gegenkathete relative to angle A and b the Ankathete relative to angle A. The term Gegenkathete is primarily used in German-language mathematics; in English, the corresponding concept is simply the opposite side relative to the angle.

Example: In a right triangle with hypotenuse length 10 and angle α with sin α = 0.6, the Gegenkathete has length a = sin α × Hypotenuse = 6. The Ankathete then has length b = cos α × Hypotenuse = 8.

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