Question

Prove law of cosine for plane angle.

Answer 1

Law of Cosines:

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. It states that, if the length of two sides and the angle between them is known for a triangle, then we can determine the length of the third side. It is given by:

c² = a² + b² – 2ab cosγ

In the right triangle BCD, by the definition of cosine function:

cos C = CD/a

or

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

or

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.