The law of cosines is one of the trigonometric laws which is used to find the angles or sides of a triangle. Once all the three sides of the triangle(SSS) are given, the angles can be easily computed using the cos law. Similarly, cosine laws can be used to calculate the sides of the triangle, where the values of angles will be given already.
Law of Cosines Definition
In Trigonometry, the law of Cosines, also known as Cosine Rule or Cosine Formula basically relates the length of the triangle to the cosines of one of its angles.
Law of Cosines Formula
As per the cosines law formula, to find the length of sides of triangle say △ABC, we can write as;
- a2 = b2 + c2– 2bc cos (A)
- b2 = a2 + c2 – 2ac cos (B)
- c2 = a2 + b2 – 2bc cos (C)
And if we want to find the angles of △ABC, then the cosine rule is applied as;
- cos A= (b2 + c2 – a2)/2bc
- cos B = (a2 + c2– b2)/2ac
- cos C = (a2 + b2– c2)/2ab
Where a, b and c are the lengths of legs of a triangle.
Summary for Cos Law Formulas:
|To Find Length||To Find Angles|
a2 = b2 + c2– 2bc cos(A)
cos A= (b2 + c2 – a2)/2bc
|b2 = a2 + c2 – 2ac cos(B)||
cos B = (a2 + c2– b2)/2ac
c2 = a2 + b2 – 2bc cos(C)
cos C = (a2 + b2– c2)/2ab
Cosines Law Proof
Now let us learn the law of cosines proof here;
In the right triangle BCD, by the definition of cosine function:
cos C = CD/a
CD=a cos C
Subtracting above equation from side b, we get
DA = b − acosC ……(1)
In the triangle BCD, according to Sine definition
sin C = BD/a
BD = a sinC ……(2)
In the triangle ADB, if we apply the Pythagorean Theorem, then
c2 = BD2 + DA2
Substituting for BD and DA from equations (1) and (2)
c2 = (a sin C)2 + (b-acosC)2
By Cross Multiplication we get:
c2 = a2 sin2C + b2 – 2abcosC + a2 cos2C
Rearranging the above equation:
c2 = a2 sin2C + a2 cos2C + b2 – 2ab cosC
Taking out a2 as a common factor, we get;
c2 = a2(sin2C + cos2C) + b2 – 2ab cosC
Now from the above equation, you know that,
sin2θ + cos2θ = 1
∴ c2 = a2 + b2 – 2ab cosC
Hence, the cosine law is proved.
Law of Cosines Problems and Solutions
Let us understand the concept by solving one of the cosines law problems.
Problem: A triangle ABC has sides a=10cm, b=7cm and c=5cm. Now, find its angle ‘x’.
Consider the below triangle as triangle ABC, where,
By using cosines law,
a2 = b2 + c2 – 2bc cos(x)
cos x = (b2 + c2 – a2)/2bc
Substituting the value of the sides of the triangle, a,b and c, we get
cos(x) = (72 + 52 – 102)/(2 × 7 × 5)
cos(x)=(49 + 25 -100)/70
It is important to solve more problems based on cosines law formula by changing the values of sides a, b & c and cross-check law of cosines calculator available here.
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