Chapter 23

Non-right-angled triangles

and some practical

applications

23.1 The sine and cosine rules

To ‘solve a triangle’ means ‘to find the values of
unknown sides and angles’. If a triangle isright-angled,
trigonometric ratios and the theorem of Pythagoras
may be used for its solution, as shown in Chapter 21.
However, for a non-right-angled triangle, trigono-
metric ratios and Pythagoras’ theorem cannot be used.
Instead, two rules, called thesine ruleand thecosine
rule,areused.

23.1.1 The sine rule

With reference to triangleABCof Figure 23.1, thesine
rulestates

a

sinA

=

b

sinB

=

c

sinC

a

c b

B C

A

Figure 23.1

The rule may be used only when

(a) 1 side and any 2 angles are initially given, or

(b) 2 sides and an angle (not the included angle) are

initially given.

23.1.2 The cosine rule

With reference to triangleABCof Figure 23.1, the

cosine rulestates

a^2 =b^2 +c^2 − 2 bccosA

or b^2 =a^2 +c^2 − 2 accosB

or c^2 =a^2 +b^2 − 2 abcosC

The rule may be used only when

(a) 2 sides and the included angle are initially

given, or

(b) 3 sides are initially given.

23.2 Area of any triangle

Thearea of any trianglesuch asABCof Figure 23.1

is given by

(a)

1

2

×base×perpendicular height

DOI: 10.1016/B978-1-85617-697-2.00023-5