Law of Cosines 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 law of cosine for angle A is given as:
a2 = b2 + c2 – 2bc cos A
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.
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.
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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
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Why is the Law of Cosines Important?
The Law of Cosines is a powerful tool for solving triangles that are not right-angled. In particular, it helps in situations where the Law of Sines may not be applicable, such as:
- SAS (Side-Angle-Side): You know two sides and the included angle.
- SSS (Side-Side-Side): You know all three sides and need to find an angle.
Real-World Applications of the Law of Cosines
- Navigation and Surveying: It’s used in determining distances and angles in land surveys.
- Physics and Engineering: In mechanics, it helps calculate forces acting on an object at various angles.
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 and Cosine 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
Common difference between sine and cosine rule are:
Feature | Law of Cosines | |
|---|---|---|
Applicability | Suitable for solving triangles where you know: 2 sides and the included angle OR 3 sides. | Suitable for solving triangles where you know: 2 angles and 1 side OR 2 sides and the included angle. |
Formula |
| a / Sin A = b/ Sin B = c / Sin C OR a : b : c = Sin A : Sin B : Sin C |
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Solved 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)
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.
Conclusion
The Law of Cosines is a versatile and essential tool in trigonometry. Whether you’re solving for sides or angles, its wide range of applications makes it an important concept to understand for anyone working with triangles. By mastering this formula, you’ll have a deeper insight into how triangles function, both in theory and in practice.