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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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 law of cosine for angle A is given as:
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.
Table of Content
- What is Law of Cosines?
- Law of Cosines Formula
- Law of Cosines Proof
- How to Find Angle using Law of Cosines
- Sine Formula
- Examples Using Law of Cosines
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