Cosine rule

When working non-right angled triangles, sometimes you may notice that there is not enough information given to use the sine rule. In this situation you can try using something called the Cosine Rule. The cosine rule states that:

c2 = a2 + b2 - 2ab cos C (Fig. 1.0)

At first, this may look scary, but it is not that bad a actually. The labels a, b, c  (all lowercase) are means the same as in the sine rule, which are the labels of the sides. Capital C is the label of the C angle, however, that can be changed to the angle you have, or the angle you are working with. The formula is completely rearrangeable.

This is a tip from a qualified IB Math Teacher. If you get an exam question which is about finding lengths, or angles of non-right angled triangles, it is highly recommended to try using the formula of the Sine Rule first, as it can be applied much quicker and easier than the cosine rule, which takes slightly more time. If it doesn't work, proceed to using the cosine rule.

(See Figure 1.1 for task)

a) 

We can find side AC easily using the cosine rule. Let AC be c, let AB be a, let BC be b.

c2 = a2 + b2 - 2ab cos C

Since the given angle is B and not C, we can change up the formula:

c2 = a2 + b2 - 2ab cos B

Substitute values:

c2 = 62 + 102 - 2 × 6 × 10 × cos 100

c2  will be about:

c2 = 156.83

Use square root to find c:

c = √ (156.83) = 12.52 cm

b)

This question wants us to find angle C. Start the same formula but keep C.

c2 = a2 + b2 - 2ab cos C

Then rearrange to get the second formula (fig. 1.0) where:

cos C =( a2 +b2 - c2 )/2ab

Substitute what you are given:

cos C =((12.52)2+ 62 -102)/2 × 6 × 10

cos C = -0.17

Then use the inverse cosine to find angle C:

C = cos-1 (-0.17)

C = 99.95°