‘positive direction’ – as shown. Note that for any angle,θ,|sinθ|and|cosθ|are both
≤1.Thenegativeofθ,−θ, means a rotation through angleθin the clockwise direction
from the positivex-axis. From Figure 6.3 it then follows that cosθis anevenfunction of
θ,cos(−θ)=cosθ and sinθ is anoddfunction ofθ,sin(−θ)=−sinθ (94
➤
). So, is
tanθeven or odd?
The signs of the trig ratios in the different quadrants can be remembered from ‘All Silly
Tom Cats’ (or you might have learnt ‘CAST’, which is not so jolly), going anticlockwise
round the diagram below
Sine only positive All positive
Tan only positive Cosine only positivewhich tells us which ratios are positive in the respective quadrants.
It is useful to memorise the trig ratios for the commonly occurring angles 30°,45°,60°,
which can be conveniently obtained from the triangles shown in Figure 6.4.
211160 °30 °45 °√ (^3) √ 2
Figure 6.430 – 60 and 45 triangles.
From Figure 6.4 we see directly that, for example sin 60°=
√
3
2
, cos 60°=
1
2
,tan30°=
1
√
3
.
The term ‘cosine’ is not accidental – cosine ofθ is the sine of( 90 −θ). As noted in
Section 5.2.2, if two anglesα,β, sum to 90°they are said to becomplementary.βis
thecomplementofαand vice versa. From the triangle shown in Figure 6.5 it is readily
C B
A
a
b
Figure 6.5αandβare complementary angles.