Transcript Thinking Mathematically by Robert Blitzer

Trigonometric
Functions: The Unit
Circle
4.2
Unit Circle
• The unit circle is a circle of radius 1 with its
center at the origin.
Definitions of the Trigonometric
Functions in Terms of a Unit Circle
If t is a real number and P = (x, y) is a point
on the unit circle that corresponds to t, then
sin t  y
cos t  x
1
csc t  , y  0
y
1
sec t  ,x  0
x
y
tan t  , x  0
x
x
cot t  , y  0
y
Points on the Unit Circle
Example
Use the Figure to find the values of the
trigonometric functions at t=/2.
Solution:
(0,1)
The point P on the unit circle that
Corresponds to t= /2 has coordinates
(0,1). We use x=0 and y=1 to find the
Values of the trigonometric functions


sin
csc
2

2
 y 1

tan t 
cos
2
x0
1 1
 x 0
  1 cot    0
y 1
2 y 1
y
1 1
and sec t    undefined
x
x 0
/2
/2
(1,0)
2
2
x  y 1
The Domain and Range of the
Sine and Cosine Functions
• The domain of the sine function and the
cosine function is the set of all real numbers
• The range of these functions is the set of all
real numbers from -1 to 1, inclusive.
Evaluating Trigonometric Functions
• Evaluate the six trig functions at each real
number.
(a) t=л/6
(b) t=5л/4
(c) t=0
(d) t=л
Evaluate the 6 Trig Functions at
t=-л/3
Even and Odd Trigonometric
Functions
The cosine and secant functions are even.
cos(-t) = cos t
sec(-t) = sec t
The sine, cosecant, tangent, and cotangent
functions are odd.
sin(-t) = -sin t
csc(-t) = -csc t
tan(-t) = -tan t
cot(-t) = -cot t
Example
• If sin t = 2/5 and cos t = 21/5, find the
remaining four trig functions
Definition of a Periodic Function
A function f is periodic if there exists a
positive number p such that
f(t + p) = f(t)
For all t in the domain of f. The smallest
number p for which f is periodic is called
the period of f.
Periodic Properties of the Sine
and Cosine Functions
sin(t + 2) = sin t and cos(t + 2) = cos t
The sine and cosine functions are periodic
functions and have period 2.
sin  = sin 3
Periodic Properties of the
Tangent and Cotangent Functions
tan(t + ) = tan t and cot(t + ) = cot t
The tangent and cotangent functions are
periodic functions and have period .
tan  = tan 2