Law of Cosines Calculator

How does this law of cosines calculator work?

Together with the law of sines, the law of cosines can help in solving from simple to complex trigonometric problems by using the formulas provided below. These calculations can be either made by hand or by using this law of cosines calculator.

A = cos^-1[(b²+c²-a²)/2bc]

Considering that a, b and c are the 3 sides of the triangle opposite to the angles A, B and C as presented within the following figure, the law of cosines states that:

■ In order to solve for the three sides (a, b and c) you should be using these equations:

a² = b² + c² - 2bc*cos(A)

a = √[b² + c² - 2bc*cos(A)]

b² = a² + c² - 2ac*cos(B)

b = √[a² + c² - 2ac*cos(B)]

c² = a² + b² - 2ac*cos(C)

c = √[a² + b² - 2ac*cos(C)]

■ In order to determine the three angles (A, B and C) you should be applying these formulas:

A = cos^-1 [(b² + c² - a²)/2bc]

B = cos^-1 [(a² + c² - b²)/2ac]

C = cos^-1 [(a² + b² - c²)/2ab]

Triangle formulas

In trigonometry the most often used triangle formulas are:

Triangle perimeter (P) = a + b + c

Triangle semi-perimeter (s) = 0.5 * (a + b + c)

Triangle area by Heron formula (A_S) = √[ s*(s - a)*(s - b)*(s - c)]

Radius of inscribed circle in the triangle (r) = √[ (s - a)*(s - b)*(s - c) / s ]

Radius of circumscribed circle around triangle (R) = (abc) / (4A_S)

Where:

a = Side a

b = Side b

c = Side c

A = Angle A

B = Angle B

C = Angle C

P = Perimeter

s = Semi-perimeter

S_A = Area

r = radius of inscribed circle

R = radius of circumscribed circle