Download Periodic Functions

Trigonometric functions
Periodic behaviour
Any function is called periodic if it
“repeats” itself on intervals of any fixed
length. For example the sine curve.
Periodicity may be defined symbolically:
A function f is periodic if there is a
positive number p such that
f (x+p) = f(x) for every x in the
domain of f.
the smallest value of p is the period of the function
Periodic behaviour in nature
wave motion: light, sound
tides: cyclic rise and
fall of seawater
water waves
seismic tremors
Periodic functions
Periodic motion
Motion that repeats itself over and over
is called Periodic Motion or Oscillation.
It always has a stable equilibrium position
1 complete
revolution
1period
1period
Periodic Function
Amplitude is the maximum multitude of displacement
from equilibrium. It is always positive.
A = max value - min value
2
Period is the the time to complete one cycle.
Cycle is one complete round trip from A to -A then
back to A.
Periodic Function
Axis of the curve is the horizontal line that is half way
between the maximum and minimum values of the
periodic curve.
y = maximum value + minimum value
2
Periodic behaviour in physics
common back-and-forth
motion of a pendulum
motion of a spring
and a block
bouncing ball
circular motion
Periodic behaviour in life
radio waves
clock mechanism
repeated steps of a
dancer
ballet Don Quixote
(32 fouette turns)
music
Questions:
Determine the period and
the amplitude of the
functions
Trig ratios of any angle
sin  =
y
P (x, y)
cos  =
 y
0
x
y
( x 2 + y 2)
x
( x 2 + y 2)
x
tan  =
y
x
Trig ratios of any angle
sin  =
y
(x 2 + y 2) positive
cos  =
P (x, y)
y
y

x
0
x
tan  =
x
( x 2 + y 2) negative
y
x
negative
all ratios are positive
only sin is positive
II
S
A
I
III
T
C
IV
only tan is positive
only cos is positive
Special angles
Special angles: 30o, 60o, 45o, 90o.
Two special triangles can be used to find the exact
values of the sine, cos, tan of special angles.
45o
30o
2
3
60o
1
2
1
45o
1
Radians and angle measure
1 radian is the measure of the angle
subtended at the centre of a circle by
arc equal to the radius of the circle
r

r
 = 1 radian
a=*r
r=a/
 =a/r
angle in radians
Graph of f(x) = sin x
1 Domain : all real numbers, R.
2 Range :
-1  y  1 .
Graph of f(x) = cos x
1 Domain : all real numbers, R.
2 Range :
-1  y  1 .
Graph of f(x) = tan x
1 Domain : all real numbers, R, x  /2  odd number.
2 Range :
all real numbers, R .
Stretches of periodic functions
2
1.5
hx = 2sinx
1
fx = sinx
0.5
sx = 0.5sinx
2
4
6
180
8
2

-0.5
360
-1
-1.5
f (x) = sin (x)
-2
period = 2 = 360
amplitude = 1
-2.5
f (x) = 2 sin (x)
f (x) = 0.5 sin (x)
period = 2 = 360
period = 2 = 360
amplitude = 2
amplitude = 0.5
10
12
Stretches of periodic functions
4
3
2
hx = sinx
1
-2
2
-1
qx = sin2x
4
6
8
180
360

2
10
12
360
rx = sin0.5x
-2
f (x) = sin (x)
-3
-4
-5
period = 2 = 360
amplitude = 1
f (x) = sin (2 x)
f (x) = sin (0.5 x)
period =  = 180
period = 4 = 720
amplitude = 1
amplitude = 1
2
Translations of periodic functions
f x = sin  x +b
g x = sin  x
h x = sin  x-c
4
vertical
2
-5
5
-2
horizontal
-4
-6