In trigonometry

Trigonometry

Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...

, the law of cotangents relates the radius of the inscribed circle

Circle

A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

 of a triangle

Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....

 to its sides and angles.

When a, b, and c are the lengths of the three sides of the triangle, and α, β, and γ are the angles opposite those three respective sides. The law of cotangent

Trigonometric function

In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...

s states that if

(the radius of the inscribed circle for the triangle) and

(the semi-perimeter for the triangle),

then the following all form the law of cotangents:

It follows that

In words the theorem is: the cotangent of a half-angle equals the ratio of the semi-perimeter minus the opposite side to the said angle, to the inradius for the triangle.

See also

• Law of sines
  Law of sines
  In trigonometry, the law of sines is an equation relating the lengths of the sides of an arbitrary triangle to the sines of its angles...
  
  
• Law of cosines
  Law of cosines
  In trigonometry, the law of cosines relates the lengths of the sides of a plane triangle to the cosine of one of its angles. Using notation as in Fig...
  
  
• Law of tangents
  Law of tangents
  In trigonometry, the law of tangents is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposite sides....
  
  
• Mollweide's formula
  Mollweide's formula
  In trigonometry, Mollweide's formula, sometimes referred to in older texts as Mollweide's equations, named after Karl Mollweide, is a set of two relationships between sides and angles in a triangle...
  
  
• Tangent half-angle formula