Sullivan Microsite DigiSample

Sullivan AP˙Sullivan˙Chapter01 October 8, 2016 17:4
Sullivan
120 Chapter 1 • Limits and Continuity
x 2 y 2 1
y
1
P (cos θ, sin θ)
x 2 y 2 1
y
1
D (1, tan θ)
P
1
O
θ
C (1, 0)
1 x
1
θ
O
B (cos θ, 0)
1
C
x
1
1
Figure 43
π
(a) 0 θ
2
(b)
We see from Figure 43(b) that
area of triangle BOP ≤ area of sector COP ≤ area of triangle COD (1)
Each of these areas can be expressed in terms of θ as
• area of triangle BOP = 1 2 cos θ · sin θ area of a triangle = 1 base × height
2
• area of sector COP = θ 2
• area of triangle COD = 1 tan θ
· 1 · tan θ =
2 2
area of a sector = 1 2 r 2 θ; r = 1
area of a triangle = 1 base × height
2
So, for 0