242 Trigonometric functions in calculus Ch. 12
The functions c and s have the following properties:-
(i)For 0 ≤x≤ 360 , 0 ≤y≤ 360 ,x+y≤ 360 ,
c(x+y)=c(x)c(y)−s(x)s(y),s(x+y)=s(x)c(y)+c(x)s(y).
(ii)For 0 ≤x≤ 360 ,c(x)^2 +s(x)^2 =1.
(iii)For 0 ≤x≤ 360 ,c(x)= 1 − 2 s(^12 x)^2 ,s(x)= 2 c(^12 x)s(^12 x).
Proof. These follow immediately from the definition ofcands, and the formulae
for cos(α⊕β)and sin(α⊕β).
Definition.For0≤x≤360 and integersn≥3, let
un(x)= 2 ns
( x
2 n+^1
)
,vn(x)= 2 n
s
( x
2 n+^1
)
c
( x
2 n+^1
).
COMMENT.Whenn≥ 1 ,unis the sum of the areas of 2n+^1 non-overlapping
congruent triangles, each having its vertex at the centre of the circle/arc with radius
length 1, and each having the end-points of its base on the circle/arc.
Figure 12.3.
Whenn≥ 1 ,vnis the sum of the areas of 2nnon-overlapping congruent triangles,
each having its vertex at the centre of the circle/arc, and each having its base-line a
tangent to the circle/arc. Our reason for takingn≥3 is that then
x
2 n+^1
≤
360
16
= 22. 5 < 45.
Figure 12.4