Geometry with Trigonometry

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