Download Trigonometric Functions of Real Numbers (Sec. 6.3)

Trigonometric Functions of Real Numbers (Sec. 6.3)
Feature
Details / Examples
Definitions of the
trigonometric functions of
real numbers
The value of a trigonometric function at a real number t is its
value at an angle of _____________, provided that value
exists.
Geometric interpretation
On the unit circle (circle with radius = 1), the graph of the
equation:
x 2 + y2 = 1
€
Definitions of the
trigonometric functions in
terms of a unit circle
Example: Finding special
values of the trigonometric
functions (not limited to
Quadrant I)
If t is a real number and P(x, y) is the point on the unit circle
that corresponds to t, then:
sin t =
cos t =
tan t =
csc t =
sec t =
cot t =
(a) t = π/4:
sin π/4 =
cos π/4 =
tan π/4 =
csc π/4 =
sec π/4 =
cot π/4 =
1
(b) t = π/2:
sin π/2 =
cos π/2 =
tan π/2 =
csc π/2 =
sec π/2 =
cot π/2 =
Theorem on Repeated
Function Values for sin
and cos
If n is any integer, then
Definition of Periodic
Function
A function is periodic if there exists a positive real number k
such that
sin(t+2πn) = sin t
and
cos(t+2πn) = cos t
f(t + k) = f(t)
for every t in the domain of f. The least such number k, if it
exists, is the period of f.
Examples of “Periodic
functions” in ordinary life
The graph of sin x
Range of sin x is [-1, 1].
The graph of cos x
Range of cos x is also [-1, 1].
How does it relate to sin x?
phase difference
The graph of tan x
tan x has an unbounded range, with vertical asymptotes.
2
Formulas for negatives
Even and Odd
Trigonometric Functions
sin π/2 =
cos π/2 =
tan π/2 =
csc π/2 =
sec π/2 =
cot π/2 =
1. Even functions: cosine, secant.
2. Odd functions: sine, tangent, cotangent, cosecant.
DAB, March 2011
3