Page 38 - Calculus Demystified

CHAPTER 1
Basics
y 25

sin

cos
x
unit circle

Fig. 1.34

and
cos θ = y = cos(θ + 2π).
We say that the sine and cosine functions have period 2π: the functions repeat
themselves every 2π units.
In practice, when we calculate the trigonometric functions of an angle θ,we
reduce it by multiples of 2π so that we can consider an equivalent angle θ , called

the associated principal angle, satisfying 0 ≤ θ < 2π. For instance,
15π/2 has associated principal angle
3π/2 (since 15π/2 − 3π/2 = 3 · 2π)

and
−10π/3 has associated principal angle
2π/3 (since − 10π/3 − 2π/3 =−12π/3 =−2 · 2π).
You Try It: What are the principal angles associated with 7π,11π/2, 8π/3,
−14π/5, −16π/7?
What does the concept of angle and sine and cosine that we have presented here
have to do with the classical notion using triangles? Notice that any angle θ such
that 0 ≤ θ< π/2 has associated to it a right triangle in the first quadrant, with
vertex on the unit circle, such that the base is the segment connecting (0, 0) to (x, 0)
and the height is the segment connecting (x, 0) to (x, y). See Fig. 1.35.