Circular Functions

Here we give a more abstract, modern approach to the definition
of trigonometric functions, which does not directly involve triangles
or angles. This enables us to employ trigonometric functions even in
cases where no angles are involved
In the Cartesian coordinate system the graph of
     
is a unit circle with center at the origin and radius 1
Using this circle, we define the circular functions on a domain of real numbers.
Let
     
be an arbitrary real number and let U be the unit circle
Begin at (1,0) and go counterclockwise along the circumference of the circle if
is positive and go clockwise along the circumference of the circle if is negative,
go until the an arc length of
     
along the circumference of the circle has been covered
Let P(a,b) be the point at the terminal end of the arc


DEFINITION 1
        

The six circular functions (commonly referred to as trigonometric functions)
are then defined in terms of the coordinates of the point P

                                 

                                                        

Notice that these definitions do not involve the mention of any angles.
The definitions involve the length of the arc whose measure is , a real number,
and the coordinates a and b of the point P that is the end-point of the arc.


                                     Sine and Cosine Domain and Range
          
By Definition 1, note that the coordinates of the point P at the end of an arc of length
     ,
starting at (1,0), can be written as P(cos x, sin x).
The domains of the sine and the cosine functions, that is, the quantities for which these functions
are defined, are real numbers.
Notice that as the point P moves around the circle the values of the sine and the cosine functions
vary from –1 at the smallest to +1 at the largest;
these values comprise the range of the sine and the cosine functions.

          Domain: The set of all real numbers
          Range:       y is a real number

where or


          See Example 1, pages 456 – 457, of the textbook.


                              Periodic Functions
Imagine the point P moving around the unit circle in either direction, clockwise or counterclockwise.
Every time P covers a distance of
          
(the circumference of the unit circle), it will be back at the point where it started.
So for any real number and any integer , it will be true that
     
and

     


Functions with this kind of repetitive behavior are called periodic functions.

DEFINITION
A function is periodic if there exists a positive real number such that
          
for all in the domain of .
The smallest such positive, if it exists, is called the fundamental period
of , or simply the period of .

The sine and the cosine functions are periodic with period
          .
The other trigonometric functions also are periodic functions.
The tangent and the cotangent functions have period
              .
The secant and the cosecant functions have period
          .
We will graph these functions in sec 6.6


     See Example 2, page 458, of the textbook.


Many phenomena in nature can be modeled using trigonometric functions
because of the periodic properties of the trigonometric functions.
For example, phenomena such as light, sound, and electromagnetic waves in general;
the motion of bridges and buildings during earthquakes; the motion of planets and satellites.


                                  Basic Identities
From the definitions of the trigonometric functions, the following identities are immediate.
For any real number (restricted so that both sides of the equation are defined):

                              Reciprocal Identities
                               


                              Quotient Identities
                    

                              Identities for Negatives

                               


                              Pythagorean Identity
                          

From the definitions of the circular functions (definition 1) and that

                    

we can easily prove the identities.

          

          

          

          


In the figure below, since the terminal points of  x  and  -x  are symmetric with respect to the horizontal axis,

          

we can write:

                              

                              

                              



To prove the Pythagorean identity, note that
          
is on the unit circle
          
so that
          .

This is usually written in the form
          

    See Example 3, page 460, of the textbook.


                    Circular Functions and Trigonometric functions
The definitions of the trigonometric functions involving angle domains can be related to the circular
functions involving real number domains.
Look at the radian measure of an angle
          
opposite an arc of units on the unit circle.


For the unit circle, the angle opposite an arc of units has a radian measure of .
In other words, every real number can be associated with an arc of units on the
unit circle or a central angle of radians on the same circle.
If is positive, go counterclockwise;
if is negative, go clockwise.
Note that the point on the terminal end of the arc of units is
also on the terminal side of the angle of radians, so we can write the following relationships

          

between trigonometric functions defined with angle domains and the trigonometric functions defined
with real number domains:

                         Circular Functions                             Trigonometric Functions

                                                   

                                                 

                                         

                                        

                                      

                                     



              See Examples 4 – 5, pages 461 – 462, of the textbook.



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