Trigonometric Functions of Real Numbers;
Periodic Functions
• Use a unit circle to define trigonometric functions of real numbers.
• Recognize the domain and range of sine and cosine functions
• Use even and odd trigonometric functions
• Use periodic functions
H.Melikian/1200
Dr .Hayk Melikyan / Departmen of Mathematics and CS melikyan@nccu.edu
1

The Unit Circle

A unit circle is a circle of radius 1, with its center at the origin of a rectangular coordinate system. The equation of this circle is x
2  y
2  r
2
.
H.Melikian/1200 2

The Unit Circle
In a unit circle, the radian measure of the central angle is
equal to the length of the intercepted arc. Both are given by the same real number t.
H.Melikian/1200 3

Definition: The Value of a trigonometric function at the real number t is its value at an angle of t radians
If t is a real number and P = (x y) is a point on the unit circle that corresponds to t, then
Unit circle y
Q = (0, 1) sin t
 y csc t

1 y cos t
 x sec t

1 x tan t
 y cot t
 x x y

/2
P(x, y)
(1, 0) x x 2 + y 2 = 1
H.Melikian/1200 4

Example: Finding Values of the Trigonometric Functions
Use the figure to find the values of the trigonometric functions at t.
sin t
 y

1
2 cos t
 x tan t
 y x

3
2
1
 2 
1
3 3

2
1 3

3
3 3 3
H.Melikian/1200 5

Example: Finding Values of the Trigonometric Functions
Use the figure to find the values of the trigonometric functions at t.
csc t

1 y
1
2
2 sec t

1 x

1

2
3 3
2

2 3

2 3
3 3 3
3 cot t
 x y
 2 
1
2
3
H.Melikian/1200 6

H.Melikian/1200 7

Text Example
Use the figure at the right to find the values of the trigonometric functions at t =

/2.
y
Solution 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.
P = (0, 1)

/2
(1, 0) x sin
 csc
2

2
 y  1, cos

2

1 1 y

1
 1, cot
 x  0

2
 x y

0
1
 0 x 2 + y 2 = 1
By definition, tan t = y / x and sec t = 1 / x
. Because x = 0, tan

/2 and sec

/2 are undefined.
H.Melikian/1200 8

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.
H.Melikian/1200 9

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 tan (-t) = – tan t csc (-t) = – csc t cot (-t) = – cot t
H.Melikian/1200 10

Example

Find the exact value of sin (-45º) sin(

45 o
)
  sin( 45 o
)
 
2
2
H.Melikian/1200 11

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.
H.Melikian/1200 12

Periodic Properties of the Sine and Cosine Functions
The sine and cosine functions are periodic functions and have period 2  .
• sin (t + 2  ) = sin t
• cos (t + 2  ) = cos t
Example
Find the exact value of cos(5  ) cos( 5

)
 cos( 4
  
)
 cos
  
1
H.Melikian/1200 13

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  .
H.Melikian/1200 14

Repetitive Behavior of the Sine, Cosine, and
Tangent Functions
For any integer n and real number t, is sin (t + 2  n) = sin t, cos (t + 2  n) = cos t, and tan (t +  n) = tan t.
H.Melikian/1200 15

Home Work
1.
H.Melikian/1200 16