Transcript Periodic Functions - Morgan Park High School

Chapter 7: Trigonometric Functions
L7.4 & 5: Graphing the Trigonometric Functions
(Part 2)
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
Unit 6
Periodic Functions
A periodic function is any function that has a continuously
repeating pattern.
A cycle is a portion of the graph that is the smallest
complete repeating pattern in the graph.
The period of a function is the length of one complete
cycle.
The axis of the curve is the horizontal line drawn halfway
between the maximum and minimum values.
The amplitude of the curve is the vertical distance from the
axis to either the maximum or minimum.
amplitude
period
3
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
Unit 6
Periodic Functions
Periodic
P = 1/60
6
Not
Periodic
Amplitude is
decreasing
Periodic
P=3
Periodic
P=2
Not
Periodic
No pattern
Periodic
P = 10
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
The sine function
Imagine a particle on the unit circle, starting at (1,0) and rotating
counterclockwise around the origin. Every position of the particle
corresponds with an angle, θ, where y = sin θ. As the particle moves
through the four quadrants, we get four pieces of the sin graph:
I. From 0° to 90° the y-coordinate increases from 0 to 1
II. From 90° to 180° the y-coordinate decreases from 1 to 0
III. From 180° to 270° the y-coordinate decreases from 0 to −1
IV. From 270° to 360° the y-coordinate increases from −1 to 0
sin θ
y
90°
135°
45°
II
I
I
0°
180°
II I
II
x
0
90°
180°
IV
225°
III
315°
270°
Interactive Sine Unwrap
360° θ
270°
IV
θ
sin θ
0
0
π/2
1
π
0
3π/2
−1
2π
0
Sine is a periodic function: p = 2π
sin θ
−3π
−2π
−π
0
π
2π
One period
2π
sin θ: Domain (angle measures): all real numbers, (−∞, ∞)
Range (ratio of sides): −1 to 1, inclusive [−1, 1]
sin θ is an odd function; it is symmetric wrt the origin.
  Domain, sin(−θ) = −sin(θ)
3π θ
The cosine function
Imagine a particle on the unit circle, starting at (1,0) and rotating
counterclockwise around the origin. Every position of the particle
corresponds with an angle, θ, where x = cos θ. As the particle moves
through the four quadrants, we get four pieces of the cos graph:
I. From 0° to 90° the x-coordinate decreases from 1 to 0
II. From 90° to 180° the x-coordinate decreases from 0 to −1
III. From 180° to 270° the x-coordinate increases from −1 to 0
IV. From 270° to 360° the x-coordinate increases from 0 to 1
y
cos θ
90°
135°
45°
II
I
I
0°
180°
II I
IV
225°
315°
270°
IV
x
0
90°
270°
180°
II
III
θ
360°
θ
cos θ
0
1
π/2
0
π
−1
3π/2
0
2π
1
Cosine is a periodic function: p = 2π
cos θ
θ
−3π
−2π
−π
0
π
One period
2π
cos θ: Domain (angle measures): all real numbers, (−∞, ∞)
Range (ratio of sides): −1 to 1, inclusive [−1, 1]
cos θ is an even function; it is symmetric wrt the y-axis.
  Domain, cos(−θ) = cos(θ)
2π
3π
Tangent Function
Recall that tan  
sin 
.
cos 
Since cos θ is in the denominator, when cos θ = 0, tan θ is undefined.
This occurs @ π intervals, offset by π/2: { … −π/2, π/2, 3π/2, 5π/2, … }
Let’s create an x/y table from θ = −π/2 to θ = π/2 (one π interval),
with 5 input angle values.
θ
sin θ
cos θ
tan θ
θ
tan θ
−π/2
−1
0
−∞
−π/2
−∞
2
2
2
2
−1
−π/4
−1
0
0
1
0
0
0
π/4
2
2
2
2
1
π/4
1
π/2
1
0
∞
π/2
∞
−π/4

Graph of Tangent Function: Periodic
tan θ
Vertical asymptotes
where cos θ = 0
tan  
θ
tan θ
−π/2
−∞
−π/4
−1
0
0
π/4
1
π/2
∞
−3π/2
−π/2
0
π/2
One period: π
tan θ: Domain (angle measures): θ ≠ π/2 + πn
Range (ratio of sides): all real numbers (−∞, ∞)
tan θ is an odd function; it is symmetric wrt the origin.
  Domain, tan(−θ) = −tan(θ)
3π/2
θ
sin 
cos 
Cotangent Function
cos 
.
sin
Since sin θ is in the denominator, when sin θ = 0, cot θ is undefined.
Recall that cot  
This occurs @ π intervals, starting at 0: { … −π, 0, π, 2π, … }
Let’s create an x/y table from θ = 0 to θ = π (one π interval),
with 5 input angle values.
cos θ
cot θ
θ
cot θ
0
1
∞
0
∞
π/4
2
2
2
2
1
π/4
1
π/2
1
0
0
π/2
0
3π/4
2
2
2
2
−1
3π/4
−1
–1
−∞
π
−∞
θ
0
π
sin θ
0

Graph of Cotangent Function: Periodic
Vertical asymptotes
where sin θ = 0
cos 
cot  
sin
cot θ
θ
tan θ
0
∞
π/4
1
π/2
0
3π/4
−1
π
−∞
−3π/2
-π
−π/2
π/2
cot θ: Domain (angle measures): θ ≠ πn
Range (ratio of sides): all real numbers (−∞, ∞)
cot θ is an odd function; it is symmetric wrt the origin.
  Domain, tan(−θ) = −tan(θ)
π
3π/2
Cosecant is the reciprocal of sine
Vertical asymptotes
where sin θ = 0
csc θ
−3π
θ
0
−2π
−π
π
2π
3π
sin θ
One period: 2π
sin θ: Domain: (−∞, ∞)
Range: [−1, 1]
csc θ: Domain: θ ≠ πn
(where sin θ = 0)
Range: |csc θ| ≥ 1
or (−∞, −1] U [1, ∞]
sin θ and csc θ
are odd
(symm wrt origin)
Secant is the reciprocal of cosine
Vertical asymptotes
where cos θ = 0
sec θ
θ
−3π
−2π
−π
0
π
2π
3π
cos θ
One period: 2π
cos θ: Domain: (−∞, ∞) sec θ: Domain: θ ≠ π/2 + πn
(where cos θ = 0)
Range: [−1, 1]
Range: |sec θ | ≥ 1
or (−∞, −1] U [1, ∞]
cos θ and sec θ
are even
(symm wrt y-axis)
Summary of Graph Characteristics
Def’n
∆
sin θ
csc θ
cos θ
sec θ
tan θ
cot θ
о
Period
Domain
Range
Even/Odd
Summary of Graph Characteristics
Def’n
Period
Domain
Range
Even/Odd
−1 ≤ x ≤ 1 or
[−1, 1]
odd
∆
о
sin θ
opp
hyp
y
r
2π
(−∞, ∞)
csc θ
1
.sinθ
r
.y
2π
θ ≠ πn
cos θ
adj
hyp
x
r
2π
(−∞, ∞)
sec θ
1 .
sinθ
r
y
2π
θ ≠ π2 +πn
tan θ
sinθ
cosθ
y
x
π
θ ≠ π2 +πn
All Reals or
(−∞, ∞)
odd
cot θ
cosθ
.sinθ
x
y
π
θ ≠ πn
All Reals or
(−∞, ∞)
odd
|csc θ| ≥ 1 or
(−∞, −1] U [1, ∞)
All Reals or
(−∞, ∞)
|sec θ| ≥ 1 or
(−∞, −1] U [1, ∞)
odd
even
even