In a non-right angled triangle, the cosine rule states that –
- b2 = a2 + c2 – 2ac cosB
- a2 = b2 + c2 – 2bc cosA
- c2 = a2 + b2 – 2ab cosC
We’ll deduct the proof of the first of the above rules below. In the triangle ABC, draw the perpendicular AD to BC, and we get two right angled triangles ABD and ADC.
In ABD, applying the Pythagoras Theorem, c2 = h2 + x2, and
sinB = and cosB =
Therefore h = c sinB, and x = c cosB
h2 = c2 – x2 – Equation 1
From ADC, we can obtain
h2 = b2 – (a – x)2
h2 = b2 – a2 + 2ax – x2 – Equation 2
Using equations 1 and 2,
h2 = c2 – x2 = b2 – a2 + 2ax – x2
c2 – x2 = b2 – a2 + 2ax – x2
c2 = b2 – a2 + 2ax
Substituting x = c cosB,
c2 = b2 – a2 + 2ac cosB
-b2 = -c2 – a2 + 2ac cosB
Therefore b2 = a2 + c2 – 2ac cosB
Similarly we can prove that a2 = b2 + c2 – 2bc cosA, and c2 = a2 + b2 – 2ab cosC.
Remember: We use cosine rule when all three sides and an included angle is given or is involved.
Now, let us look at some examples of using cosine rule
Example 1: In the triangle ABC, a = 5 cm, c = 6 cm, and angle B = 40°. What is x?
Applying the cosine rule b2 = a2 + c2 – 2ac cosB
x2 = 52 + 62 – 2 x 5 x 6 x cos40°
= 25 + 36 – 60 cos40°
x = 3.88 cm
Example 2: In the triangle ABC, a = 6 cm, b = 5 cm, c = 3 cm. What is angle A?
Applying the cosine rule a2 = b2 + c2 – 2bc cosA, and rearranging the formula, we get
= 97°40′ (to the nearest minute)
Summary of Cosine Rule
- a2 = b2 + c2 – 2bc cosA or cosA =
- b2 = a2 + c2 – 2ac cosB or cosB =
- c2 = a2 + b2 – 2ab cosC or cosC =