Cosine Theorem

Cosine theorem is similar to the sine theorem. Cosine theorem is based on Pythagorean Theorem. Cosine theorem is derived from right angled triangle.

Equation for cosine theorem is: c² = a²+b²-2abCosC

Where a,b,c are constants and C is the degree value.

Types of Cosine Theorem

There are three types of cosine theorems:

• Theorem of plane trigonometry.
• Angle cosine theorem or rule of spherical trigonometry.
• Cosine theorem or rule for sides

The cosine theorem of plane trigonometry:

Below statement states if a, b, c is sides and α, β, γ is angles of a triangle, then any one side square is equal to addition of squares of other two sides less twice the product of those sides and the cosine of the angle they include.

a² = b² + c ² - 2bc cos α
b² = c² + a ² - 2ca cos β
c² = a² + b ² - 2ab cos γ

Plane trigonometry theorem used to solve a triangle if the three side value of triangle is given or two sides and included angles are given.

Angle cosine theorem (rule) of spherical trigonometry

cos α = - cos β cos γ ² + sin β sin γ cos a
cos β = - cos γ cos α² + sin γ sin α cos b
cos γ = - cos α cos β² + sin α sin β cos c

Above statement states if a, b, and c is the sides and α, β and γ is angles of a spherical triangle.

Cosine theorem (rule) for sides of spherical triangle

Below statement states if a, b, c is sides and α, β, γ is angles of a spherical triangle, then
angle cosine rule can be used if three angles of spherical triangle are given or one side and two adjacent angles are given.

cos a = - cos b cos c² + sin b sin c cos α
cos b = - cos c cos a² + sin c sin a cos β
cos c = - cos a cos b² + sin a sin b cos γ

Cosine rule for sides of spherical triangle can be used if three sides or two sides and the included angle are given.