Understanding Engineering Mathematics

These results may be used to solve triangles given appropriate information. A triangle can
be ‘solved’ for all angles and sides if we know either:

• three sides


• two sides and an included angle


• two angles and one side


• two sides and a non-included angle – but in this case the answers can
  be ambiguous (see RE6.3.3(ii)).

Solution to review question 6.1.3

This is the case of two sides and an included acute angle.

(i) By the cosine rule, withb=2,c=5andA= 60 °we have

a^2 = 22 + 52 − 2 × 2 × 5 ×cos 60°

= 4 + 25 − 2 × 10 ×

1

2

= 19

So

a=

√

19

(ii) We can now find sinθfrom the sine rule

a

sin 60°

=

√

19

sin 60°

=

2

sinθ

So

sinθ=

2sin60°

√

19

=

√

3

√

19

= 0 .3974 to 4 decimal places

Henceθ∼= 23. 4 °.

Note: The other angle is obtuse (180°− 23. 4 °), but the sine rule

applied to this angle would not tell us this – it is always safest to

go for the angle that is obviously acute when using the sine rule to

solve triangles.

6.2.4 Graphs of the trigonometric functions

➤

172 195➤

We now focus on the trig ratios asfunctions(90
➤
) and look at their graphs (91
➤
).
This requires careful thought about what happens to the trig functions as the independent
variable,θ, changes. As the trig functions are functions of an angular variable their values
will keep repeating – because, for example,θ+ 360 °is in the same angular position asθ,
and therefore sin(θ+ 360 °)=sinθ,cos(θ+ 360 °)=cosθ, etc. We express this generally
by saying that the trigonometric functions cos, sin, and tan areperiodic with period 360°
(or 2p)by which we mean that for such functions (NB: in future we will usually use
radian measure for angles, rather than degrees – it is by far the safest policy when we are
regarding the trig ratios as functions, particularly in calculus – as a rule of thumb, if a