Page 39 - Calculus Demystified

CHAPTER 1
26
y Basics
unit circle
opposite (x, y)
side

x

adjacent side

Fig. 1.35

Then
y opposite side of triangle
sin θ = y = =
1 hypotenuse
and
x adjacent side of triangle
cos θ = x = = .
1 hypotenuse
Thus, for angles θ between 0 and π/2, the new definition of sine and cosine using
the unit circle is clearly equivalent to the classical definition using adjacent and
opposite sides and the hypotenuse. For other angles θ, the classical approach is to
reduce to this special case by subtracting multiples of π/2. Our approach using the
unit circle is considerably clearer because it makes the signatures of sine and cosine
obvious.
Besides sine and cosine, there are four other trigonometric functions:

y sin θ
tan θ = =
x cos θ
x cos θ
cot θ = =
y sin θ
1 1
sec θ = =
x cos θ
1 1
csc θ = = .
y sin θ
Whereas sine and cosine have domain the entire real line, we notice that tan θ and
sec θ are undefined at odd multiples of π/2 (because cosine will vanish there) and
cot θ and csc θ are undefined at even multiples of π/2 (because sine will vanish
there). The graphs of the six trigonometric functions are shown in Fig. 1.36.

EXAMPLE 1.19
Compute all the trigonometric functionsfor the angle θ = 11π/4.