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.