ABSTRACT
Consider a point P on the circle of radius r shown in Figure 3.2. The
angle θ between the line from the origin to P and the (positive) x-axis is
defined as the ratio of the length the arc of a circle between P and the
point (r, 0) to the radius r.1 Denoting the coordinates of P by (x, y), the
basic (circular) trigonometric functions are then defined by
cos θ = x
r , (3.2)
sin θ = y
r , (3.3)
tan θ = sin θ
cos θ , (3.4)
and the fundamental identity
cos2 θ + sin2 θ = 1 (3.5)
follows from the definition of a circle. We then have the well-known addition
formulas2
sin(θ + φ) = sin θ cosφ+ cos θ sinφ, (3.6)
cos(θ + φ) = cos θ cosφ− sin θ sinφ, (3.7) tan(θ + φ) =
tan θ + tanφ
as well as the derivative formulas3
d
dθ sin θ = cos θ, (3.9)
d
dθ cos θ = − sin θ. (3.10)
An important class of trigonometric problems involves determining, say,
cos θ if tan θ is known. One can, of course, do this algebraically by using
the identity
cos2 θ = 1
1 + tan2 θ . (3.11)
It is often easier to do this geometrically, as illustrated by the following
example.