Periodic
Functions
Sec. 4.3c
Let’s consider…
Light is refracted (bent) as it passes through glass. In the figure,
1 is the angle of incidence and  2 is the angle of refraction. The
index of refraction is a constant  that satisfies the equation
sin 1   sin 2
If 1  83 and  2  36 for a certain piece of flint glass, find
the index of refraction.
sin 83   sin 36
sin 83

 1.689
sin 36
Periodic Functions
A function
such that
y  f  t  is periodic if there is a positive number c
f  t  c   f  t  for all values of t in the domain of f.
The smallest such number c is called the period of the function.
What are some common periodic functions???
The sine, cosine, and tangent functions!!!
(what are their periods?)
Period of sine and cosine:
Period of tangent:

2
General Information about Periodic Functions
… Sin, Cos, and Tan
Periodic Function = cyclical, repeating function
Cycle = one complete pattern
Period = horizontal length of one complete pattern
Amplitude = (max-min)/2
Phase Shift = horizontal translation – what will this do to our
periodic functions?
Vertical translation – what would this do to our graphs?
Periodic Functions
Find each of the following without a calculator.
Rewrite:
 57,800 
 57,801 

1. sin 
  sin  

2
2


2


 Note: 28,900 is just a
 sin   28,900  large multiple of 2 …
2

 
 sin    1
2
Periodic Functions
Find each of the following without a calculator.
2.
cos  288.45   cos  280.45 
Rewrite:
 cos  280.45  8   cos  280.45   0


Note: 280.45 and 280.45  8 wrap to the
same point on the unit circle, so the cosine of one is
the same as the cosine of the other…
Periodic Functions
Find each of the following without a calculator.





3. tan   99,999   tan    1
4

4

Note: Since the period of the tangent function is
rather than 2 , 99,999 is a large multiple of
the period of the tangent function…
In groups of two or three, explain to each other why each of the
following statements are true. Base your explanations on the
unit circle. Remember that –t wraps the same distance as t, but
in the opposite direction.
P  cos t ,sin t 
1. For any t, the value of cos(t) lies
between –1 and 1, inclusive.
The x-coordinates on the unit circle
lie between –1 and 1, and cos(t) is
always an x-coordinate on the unit
circle.
t
t
In groups of two or three, explain to each other why each of the
following statements are true. Base your explanations on the
unit circle. Remember that –t wraps the same distance as t, but
in the opposite direction.
P  cos t ,sin t 
2. For any t, the value of sin(t) lies
between –1 and 1, inclusive.
The y-coordinates on the unit circle
lie between –1 and 1, and sin(t) is
always a y-coordinate on the unit
circle.
t
t
In groups of two or three, explain to each other why each of the
following statements are true. Base your explanations on the
unit circle. Remember that –t wraps the same distance as t, but
in the opposite direction.
P  cos t ,sin t 
3. The values of cos(t) and cos(–t)
are always equal to each other
(recall that this is the check for an
even function).
The points corresponding to t and –t
on the number line are wrapped to points
above and below the x-axis with the same
x-coordinates  cos(t) and cos(–t) are equal.
t
t
In groups of two or three, explain to each other why each of the
following statements are true. Base your explanations on the
unit circle. Remember that –t wraps the same distance as t, but
in the opposite direction.
P  cos t ,sin t 
4. The values of sin(t) and sin(–t)
are always opposites of each other
(recall that this is the check for an
odd function).
t
t
The points corresponding to t and –t
on the number line are wrapped to points
above and below the x-axis with exactly
opposite y-coordinates  sin(t) and sin(–t) are opposites.
In groups of two or three, explain to each other why each of the
following statements are true. Base your explanations on the
unit circle. Remember that –t wraps the same distance as t, but
in the opposite direction.
P  cos t ,sin t 
5. The values of sin(t) and sin(t + 2 )
are always equal to each other.

Since 2 is the distance around the
unit circle, both t and t + 2 get
wrapped to the same point.

 This is true for all six trigonometric functions!!!
t
t
In groups of two or three, explain to each other why each of the
following statements are true. Base your explanations on the
unit circle. Remember that –t wraps the same distance as t, but
in the opposite direction.
P  cos t ,sin t 
t
6. The values of sin(t) and sin(t +  )
are always opposites of each other
(the same is true of cos(t) and
cos(t +  )).
t

The points corresponding to t and t +
get wrapped to points on either end of a
diameter on the unit circle. These points are symmetric
with respect to the origin and therefore have coordinates
(x, y) and (–x, –y). Therefore sin(t) and sin(t + ) are
opposites.

In groups of two or three, explain to each other why each of the
following statements are true. Base your explanations on the
unit circle. Remember that –t wraps the same distance as t, but
in the opposite direction.
P  cos t ,sin t 
7. The values of tan(t) and tan(t +  )
are always equal to each other
(unless they are both undefined).
t
t
By our previous observation, tan(t)
and tan(t + ) are ratios of the form

y
x
and
y
,
x
which are either equal to each other or both undefined.
In groups of two or three, explain to each other why each of the
following statements are true. Base your explanations on the
unit circle. Remember that –t wraps the same distance as t, but
in the opposite direction.
P  cos t ,sin t 

8. The sum cos t
always equals 1.
   sin t 
2
t
2
The sum is always of the form
2
2
x  y  1 for some (x, y) on the
unit circle. Since the equation of
2
2
the unit circle is x  y  1,
the sum is always 1.
t
In groups of two or three, explain to each other why each of the
following statements are true. Base your explanations on the
unit circle. Remember that –t wraps the same distance as t, but
in the opposite direction.
P  cos t ,sin t 
At this point, we can use reference
triangles and quadrantal angles
to evaluate trig. functions for all
integer multiples of 30 or 45 .
This leads us to our 16-point unit
circle, which you must commit
to memory!!!
t
t