Cosine Rule

What is the Cosine Rule for a Triangle?

According to the Cosine Rule, the square of one side of a triangle, equals the sum of the squares of the other two sides, subtracted by twice the product of those two sides and the cosine of the angle between them. Mathematically it is written as: a² = b² + c² – 2bc cos γ .

When can we use the Cosine Rule?

The Cosine Rule is useful for finding: The third side of a triangle by using the lengths of two known sides and the angle formed between them. The angles of a triangle by using all three sides of triangle.

How to Find Angles using the Cosine Rule, when Three Sides of Triangle are given?

If γ is the included angle, then

cos γ = (a² + b² – c²)/2ab

Where a,b and c are the sides of the triangle.

What is the Cosine Formula?

The Cosine formula to find the side of the triangle is given by:

c = √[a² + b² – 2ab cos γ ] where a,b and c are the sides of the triangle.

What is the Sine Rule formula?

As per the sine rule, if α, β , and γ are the length of sides of a triangle and A, B, and C are the angles, then a/sin α = b/sin β = c/sin γ

Does Pythagoras’ Theorem only Work for Cosine?

Pythagoras’ theorem tells us that the square of the hypotenuse side is equal to the sum of squares of the other two sides. However, it works for right-angled triangles so that we can check whether a triangle has a right angle or not.

How to Find the Missing Side or Angle of Triangle Using Cosine Rule?

• To determine the side lengths of triangle ABC, we can express it as: a² = b² + c² – 2bc cos α, b² = a² + c² – 2ac cos β and c² = b² + a² – 2ba cos γ
• To calculate the angles of triangle ABC, the cosine rule express in the following manner: cos α = [b² + c² – a²]/2bc, cos β = [a² + c² – b2]/2ac and cos γ = [b² + a² – c²]/2ab

Does the Cosine Rule Work in All Triangles?

Yes, the Cosine Rule can be applied to all the triangles. It stands true for both right and oblique triangles.

What is the difference between Cosine and Sine?

In a right angled triangle Cosine is the ratio of its base to hypotenuse whereas the Sine is the ratio of perpendicular and hypotenuse.

Why is Cosine Rule important?

• It helps in derivation of Pythagoras theorem and triangle inequality.
• It is used to calculate the unknown length of a side and angle of triangle.

Cosine Rule commonly referred to as the Law of Cosines in Trigonometry establishes a mathematical connection involving all three sides of a triangle and one of its angles. Cosine Rule is most useful for solving the unknown information of a triangle. For example, when all three sides of a triangle are known, the Cosine Rule allows the determining of any angle measurement. Similarly, if two sides and the included angle between them are known, this rule facilitates the calculation of the third side length.

The Cosine Rule is a relationship between the lengths of a triangle’s sides and the cosine of one of its angles, allowing us to calculate distances and angles. When computing the third side of a triangle if two sides and their included angle are given, and when computing the angles of a triangle if all three sides are known, in that case, the Cosine Rule plays a valuable role.

In this article, we will discuss the introduction, definition, properties, formula of the Cosine Rule, and its meaning. We will also understand the proof of the Cosine Rule. We will also solve various examples and provide practice questions based on Cosine Rule for a better understanding of the concept of this article.