Law of Cosines

Law of Cosines in Trigonometry is the fundamental law of mathematics used to find the angle of the triangle when all three sides of the triangle are given. This law is also called the Cosine Rule Or the Cosine Formula, if in a triangle the sides are a, b, and c, then

a² = b² + c² – 2bc cos A

Similarly, all other angles B and C are given. In this article, we will learn about, the Law of Cosines, the Law of Cosines formula, examples, and others in detail.

What is Law of Cosines?

Law of cosine is the fundamental law of trigonometry the law of cosine is used to find the relation between sides and the angles of the triangle, suppose we are given sides of the triangle and then the angle of the triangle is found. This formula is used to find the unknown angle of the triangle when all sides are given.

Learn,

• Cosine Rule
• Cosine Formulas
• Law of Sine and Cosine

Law of Cosines Definition

Law of cosine is defined as the law which gives the relation between sides and angles of the triangle. Three laws of cosine are,

• a² = b² + c² – 2bc cosA
• b² = c² + a² – 2ca cosB
• c² = a² + b² – 2ab cosC

where a, b, and c are the sides of the triangle and A, B, and C are the angles of the triangle.

Law of Cosines Formula

Cosine Formula is the formula that gives the relation between the sides and the angles of the triangle. Suppose we are given a triangle in which the sides a, b, and c are given and A, B, and C are angles of the triangle respectively. Then the law of cosine formulas is,

• a² = b² + c² – 2bc cos A
• b² = a² + c² – 2ac cos B
• c² = b² + a² – 2ba cos C

We can also find the angles of the triangle by the formulas,

• cos A = [b² + c² – a²]/2bc
• cos B = [a² + c² – b²]/2ac
• cos C = [b² + a² – c²]/2ab

Law of Cosines Proof

Law of cosine is proved using trigonometric identities. Suppose we are given a triangle ABC and BM is the altitude of the triangle and its height is h and AM is equal to r. Also, sides of the triangle a, b, and c, the image for the same is added below,

In ΔABM,

sin A = BM/AB = h/c . . . (i)

cos A = AM/AB = r/c . . . (ii)

From equation (i) and (ii),

• h = c(sin A)
• r = c(cos A)

By Pythagoras Theorem in ΔBMC,

a² = h² + (b – r)²

Then,

Using h = c(sin A), r = c(cos A) in above equation

⇒ a² = {c(sinA)}² + {b – c(cosA)}²

⇒ a² = c²sin²A + b² + c²cos²A – 2bc cosA

⇒ a² = c²(sin²A + cos²A) + b² – 2bc cosA

⇒ a² = c² + b² – 2bc cosA

This is the cosine formula.

Similarly other two formulas are also proved,

• b² = c² + a² – 2ca cosB
• c² = a² + b² – 2ab cosC

How to Find Angle using Law of Cosines

Law of Cosine is used to find the angles of the triangle when all three sides of the triangle are given. Suppose we are given a triangle with sides a, b, and c and angles of triangle are A, B, and C then angles of the triangle are calculated using the formula,

• cos A = [b² + c² – a²]/2bc
• cos B = [a² + c² – b²]/2ac
• cos C = [b² + a² – c²]/2ab

Example, Find ∠A of triangle ABC, where sides of triangles, a, b, and c, are 1 cm, 1 cm, and √2 cm.

Using cosine rule,

cos A = [b² + c² – a²]/2bc

cos A = {(1)² + (√2)² – (1)²}/2(1)(√2) = 2/2√(2)

cos A = 1/√(2)

A = cos^-1(1/√(2)) = 45°

Sine Rule

For a triangle with sides a, b, and c and angles A, B, and C then the Sine Law Formula is,

a / Sin A = b/ Sin B = c / Sin C

Learn, Law of Sine

We can also represent the sine formula as,

a : b : c = Sin A : Sin B : Sin C

Examples Using Law of Cosines

Example 1: If two sides of the triangle are 12 cm and 16 cm and the angle between them is 30° then find the third side of the triangle.

Solution:

Given,

• b = 12 cm
• c = 16 cm
• ∠A = 30°

Law of Cosines Formula,

a² = b² + c² – 2bc·cosA

⇒ a² = (12)² + (16)² – 2(12)(16)cos30°

⇒ a² = 144 + 256 – (384)(1/2) = 208

⇒ a = 14.4 cm

Thus, the third side of the triangle is 14.4 cm

Example 2: If two sides of the triangle are 8 cm, 10 cm, and 6 cm then find the angle ‘A’ of the triangle.

Solution:

Given,

• a = 8 cm
• b = 10 cm
• c = 6 cm

Using Cosines Law,

a² = b² + c² – 2bc cos(A)

⇒ cos A = (b² + c² – a²)/2bc

Substituting the given value,

cos(A) = (10² + 6² – 8²)/(2 × 10 × 6)

⇒ cos(A) = (100 + 36 – 64)/120 = 72/120 = 3/5

⇒ A = cos^-1 (3/5)

Law of Cosines Practice Problems

Probelm 1: If two sides of the triangle are 20 cm and 22 cm and the angle between them is 45° then find the third side of the triangle.

Probelm 2: If two sides of the triangle are 3 cm, 4 cm, and 5 cm then find the angle ‘A’ of the triangle.

Probelm 3: If two sides of the triangle are 8 cm and 12 cm and the angle between them is 60° then find the third side of the triangle.

Probelm 4: If two sides of the triangle are 12 cm, 18 cm, and 16 cm then find the angle ‘A’ of the triangle.

Law of Cosines – FAQs

1. What is Law of Cosine in Trigonometry?

Law of Cosine in Trigonometry is the fundamental law of trigonometry that is used to find the angle of the triangle when all three sides of the triangle are given.

2. What of Law of Cosine Formula?

Law of Cosine formula is,

• a² = b² + c² – 2bc·cos A
• b² = c² + a² – 2ca·cos B
• c² = a² + b² – 2ab·cos C

3. What is Law of Cosine class 11?

Law of Cosine class 11 is the fundamental law of mathematics that is used to find the angle of the triangle when all three sides of the triangle are given.

4. Does Law of Cosines work for All Triangles?

Yes. Law of Cosine work for all triangles and it is used for finding angles of the triangle when all its sides are given.

5. When to Use Law of Cosines?

Cosine rule or cosine formula is used when all three sides of triangle are given and angle of triangle is found.

6. How to Find Angles Using Cosine Rule?

If in a triangle where a, b and c are the sides of the triangle and C is the angle included between sides, a and b then angle C is calculated as,

Cos C = (a² + b² – c²)/2ab

7. What is Sine Rule Formula?

As per the sine rule, if a, b, and c are the length of sides of a triangle and A, B, and C are the angles, then,

a/sin A = b/sin B = c/ sin C