3. The records of star-positions over the years,
the temperatures throughout the seasons, or one's
cardiac measurements are all examples of cyclic or
periodic data,
Periodic Functions
4. The records of star-positions over the years,
the temperatures throughout the seasons, or one's
cardiac measurements are all examples of cyclic or
periodic data, i.e. repetitive measurements having a
basic block appearing at a regular interval-a period.
Periodic Functions
5. The records of star-positions over the years,
the temperatures throughout the seasons, or one's
cardiac measurements are all examples of cyclic or
periodic data, i.e. repetitive measurements having a
basic block appearing at a regular interval-a period,
Periodic Functions
Given a function f(x), f(x) is periodic if there exists a
nonzero number b such that f(x) = f(x + b) for all x’s,
6. The records of star-positions over the years,
the temperatures throughout the seasons, or one's
cardiac measurements are all examples of cyclic or
periodic data, i.e. repetitive measurements having a
basic block appearing at a regular interval-a period,
Periodic Functions
Given a function f(x), f(x) is periodic if there exists a
nonzero number b such that f(x) = f(x + b) for all x’s,
and the smallest number p > 0 where f(x) = f(x + p)
is called the period of f(x).
7. The records of star-positions over the years,
the temperatures throughout the seasons, or one's
cardiac measurements are all examples of cyclic or
periodic data, i.e. repetitive measurements having a
basic block appearing at a regular interval-a period,
Periodic Functions
Given a function f(x), f(x) is periodic if there exists a
nonzero number b such that f(x) = f(x + b) for all x’s,
and the smallest number p > 0 where f(x) = f(x + p)
is called the period of f(x).
The graph of
a periodic function
8. The records of star-positions over the years,
the temperatures throughout the seasons, or one's
cardiac measurements are all examples of cyclic or
periodic data, i.e. repetitive measurements having a
basic block appearing at a regular interval-a period,
one period p
Periodic Functions
Given a function f(x), f(x) is periodic if there exists a
nonzero number b such that f(x) = f(x + b) for all x’s,
and the smallest number p > 0 where f(x) = f(x + p)
is called the period of f(x).
p0
The graph of
a periodic function
9. The records of star-positions over the years,
the temperatures throughout the seasons, or one's
cardiac measurements are all examples of cyclic or
periodic data, i.e. repetitive measurements having a
basic block appearing at a regular interval-a period,
one period p
Periodic Functions
Given a function f(x), f(x) is periodic if there exists a
nonzero number b such that f(x) = f(x + b) for all x’s,
and the smallest number p > 0 where f(x) = f(x + p)
is called the period of f(x).
x x+p
For all x’s, f(x) = f(x+p)
p
p0
The graph of
a periodic function
10. The records of star-positions over the years,
the temperatures throughout the seasons, or one's
cardiac measurements are all examples of cyclic or
periodic data, i.e. repetitive measurements having a
basic block appearing at a regular interval-a period,
one period p
Periodic Functions
Given a function f(x), f(x) is periodic if there exists a
nonzero number b such that f(x) = f(x + b) for all x’s,
and the smallest number p > 0 where f(x) = f(x + p)
is called the period of f(x).
Since the trig-functions
are defined by positions
on the unit circle, so
trig-functions are periodic
with periods 2π (or π).
x x+p
For all x’s, f(x) = f(x+p)
p
p0
The graph of
a periodic function
12. Graphs of Trig–Functions
The graph of y = sin(θ)
An ant, starting from the point (1, 0) runs counter-
clockwise around the unit circle.
(1, 0)
13. Graphs of Trig–Functions
The graph of y = sin(θ)
An ant, starting from the point (1, 0) runs counter-
clockwise around the unit circle. The arc-distance it
covered is the radian measurement of the angle θ as
shown
(1, 0)
14. Graphs of Trig–Functions
The graph of y = sin(θ)
An ant, starting from the point (1, 0) runs counter-
clockwise around the unit circle. The arc-distance it
covered is the radian measurement of the angle θ as
shown
θ
(x, y)
(1, 0)
θ
15. Graphs of Trig–Functions
The graph of y = sin(θ)
An ant, starting from the point (1, 0) runs counter-
clockwise around the unit circle. The arc-distance it
covered is the radian measurement of the angle θ as
shown
θ
(x, y)
(1, 0)
θ
θ
θ
16. Graphs of Trig–Functions
y
An ant, starting from the point (1, 0) runs counter-
clockwise around the unit circle. The arc-distance it
covered is the radian measurement of the angle θ as
shown and sin(θ) = y = the height of the ant’s position.
θ
(x, y)
(1, 0)
θ
The graph of y = sin(θ)
θ
θ
17. Graphs of Trig–Functions
The graph of y = sin(θ)
An ant, starting from the point (1, 0) runs counter-
clockwise around the unit circle. The arc-distance it
covered is the radian measurement of the angle θ as
shown and sin(θ) = y = the height of the ant’s position.
By plotting the points (θ, y=sin(θ)),
θθ
(x, y)
yy
(1, 0) θ
θ
18. Graphs of Trig–Functions
The graph of y = sin(θ)
An ant, starting from the point (1, 0) runs counter-
clockwise around the unit circle. The arc-distance it
covered is the radian measurement of the angle θ as
shown and sin(θ) = y = the height of the ant’s position.
By plotting the points (θ, y=sin(θ)),
θθ
(x, y)
yy
(1, 0) θ
θ
19. Graphs of Trig–Functions
The graph of y = sin(θ)
An ant, starting from the point (1, 0) runs counter-
clockwise around the unit circle. The arc-distance it
covered is the radian measurement of the angle θ as
shown and sin(θ) = y = the height of the ant’s position.
By plotting the points (θ, y=sin(θ)),
θθ
(x, y)
yy
(1, 0) θ
θ
20. Graphs of Trig–Functions
The graph of y = sin(θ)
An ant, starting from the point (1, 0) runs counter-
clockwise around the unit circle. The arc-distance it
covered is the radian measurement of the angle θ as
shown and sin(θ) = y = the height of the ant’s position.
By plotting the points (θ, y=sin(θ)),
θθ
π
2
(x, y)
yy
(1, 0) θ
θ
21. Graphs of Trig–Functions
The graph of y = sin(θ)
An ant, starting from the point (1, 0) runs counter-
clockwise around the unit circle. The arc-distance it
covered is the radian measurement of the angle θ as
shown and sin(θ) = y = the height of the ant’s position.
By plotting the points (θ, y=sin(θ)),
θθ
π
2
π
(x, y)
yy
(1, 0) θ
θ
22. Graphs of Trig–Functions
The graph of y = sin(θ)
An ant, starting from the point (1, 0) runs counter-
clockwise around the unit circle. The arc-distance it
covered is the radian measurement of the angle θ as
shown and sin(θ) = y = the height of the ant’s position.
By plotting the points (θ, y=sin(θ)),
θθ
π
2
π 2π
(x, y)
yy
(1, 0) θ
θ
23. Graphs of Trig–Functions
The graph of y = sin(θ)
An ant, starting from the point (1, 0) runs counter-
clockwise around the unit circle. The arc-distance it
covered is the radian measurement of the angle θ as
shown and sin(θ) = y = the height of the ant’s position.
By plotting the points (θ, y=sin(θ)),
θθ
π
2
π 2π
(x, y)
yy
(1, 0) θ
θ
24. Graphs of Trig–Functions
The graph of y = sin(θ)
An ant, starting from the point (1, 0) runs counter-
clockwise around the unit circle. The arc-distance it
covered is the radian measurement of the angle θ as
shown and sin(θ) = y = the height of the ant’s position.
By plotting the points (θ, y=sin(θ)), we obtain the
undulating sine wave as shown.
θθ
π
2
π 2π
(x, y)
yy
(1, 0) θ
θ
25. Graphs of Trig–Functions
The graph of y = sin(θ)
An ant, starting from the point (1, 0) runs counter-
clockwise around the unit circle. The arc-distance it
covered is the radian measurement of the angle θ as
shown and sin(θ) = y = the height of the ant’s position.
By plotting the points (θ, y=sin(θ)), we obtain the
undulating sine wave as shown.
Here are the important properties of the sine wave.
θθ
π
2
π 2π
(x, y)
yy
(1, 0) θ
θ
27. Properties of y = sin(θ)
1. It’s periodic with period 2π and | sin(θ) | ≤ 1 for all θ.
Graphs of Trig–Functions
0 π 2π–π–2π 3π–3π
y = 1
y = –1
28. 2. sin(θ) = 0 for θ = .., –2π, –π, 0, π, 2π.. or θ = {nπ}.
Properties of y = sin(θ)
1. It’s periodic with period 2π and | sin(θ) | ≤ 1 for all θ.
Graphs of Trig–Functions
0 π 2π–π–2π 3π–3π
sin(θ) = 0
for θ = {nπ} y = 1
y = –1
29. 2. sin(θ) = 0 for θ = .., –2π, –π, 0, π, 2π.. or θ = {nπ}.
Properties of y = sin(θ)
1. It’s periodic with period 2π and | sin(θ) | ≤ 1 for all θ.
Graphs of Trig–Functions
0 π 2π–π–2π 3π–3π
sin(θ) = 0
for θ = {nπ} y = 1
y = –1
The ant is on the x-axis
if sin(θ) = 0 and it does this
twice for every cycle.
30. 2. sin(θ) = 0 for θ = .., –2π, –π, 0, π, 2π.. or θ = {nπ}.
sin(θ) = 1 for θ = .., –7π/2, –3π/2, 1π/2, 5π/2, 9π/2..
or θ = {2nπ + π/2)}.
Properties of y = sin(θ)
1. It’s periodic with period 2π and | sin(θ) | ≤ 1 for all θ.
sin(θ) =1
for θ = {π/2+2nπ)}
Graphs of Trig–Functions
0 π 2π–π–2π 3π–3π
sin(θ) = 0
for θ = {nπ} y = 1
y = –1
31. 2. sin(θ) = 0 for θ = .., –2π, –π, 0, π, 2π.. or θ = {nπ}.
sin(θ) = 1 for θ = .., –7π/2, –3π/2, 1π/2, 5π/2, 9π/2..
or θ = {2nπ + π/2)}.
Properties of y = sin(θ)
1. It’s periodic with period 2π and | sin(θ) | ≤ 1 for all θ.
sin(θ) =1
for θ = {π/2+2nπ)}
Graphs of Trig–Functions
0 π 2π–π–2π 3π–3π
sin(θ) = 0
for θ = {nπ} y = 1
y = –1
The ant is at the apex
if sin(θ) = 1 and it does
this once every round.
32. 2. sin(θ) = 0 for θ = .., –2π, –π, 0, π, 2π.. or θ = {nπ}.
sin(θ) = 1 for θ = .., –7π/2, –3π/2, 1π/2, 5π/2, 9π/2..
or θ = {2nπ + π/2)}.
sin(θ) = –1 for θ = .., –9π/2, –5π/2, –1π/2, 3π/2, 7π/2,..
or θ = {2nπ – π/2)}.
Properties of y = sin(θ)
1. It’s periodic with period 2π and | sin(θ) | ≤ 1 for all θ.
sin(θ) =1
for θ = {π/2+2nπ)}
Graphs of Trig–Functions
0 π 2π–π–2π 3π–3π
sin(θ) = –1
for θ = {–π/2+2nπ)}
sin(θ) = 0
for θ = {nπ} y = 1
y = –1
33. 2. sin(θ) = 0 for θ = .., –2π, –π, 0, π, 2π.. or θ = {nπ}.
sin(θ) = 1 for θ = .., –7π/2, –3π/2, 1π/2, 5π/2, 9π/2..
or θ = {2nπ + π/2)}.
sin(θ) = –1 for θ = .., –9π/2, –5π/2, –1π/2, 3π/2, 7π/2,..
or θ = {2nπ – π/2)}.
Properties of y = sin(θ)
1. It’s periodic with period 2π and | sin(θ) | ≤ 1 for all θ.
sin(θ) =1
for θ = {π/2+2nπ)}
Graphs of Trig–Functions
0 π 2π–π–2π 3π–3π
sin(θ) = –1
for θ = {–π/2+2nπ)}
sin(θ) = 0
for θ = {nπ} y = 1
y = –1
The ant is at the nadir
if sin(θ) = 1 and it does
this once every round.
34. 2. sin(θ) = 0 for θ = .., –2π, –π, 0, π, 2π.. or θ = {nπ}.
sin(θ) = 1 for θ = .., –7π/2, –3π/2, 1π/2, 5π/2, 9π/2..
or θ = {2nπ + π/2)}.
sin(θ) = –1 for θ = .., –9π/2, –5π/2, –1π/2, 3π/2, 7π/2,..
or θ = {2nπ – π/2)}.
Properties of y = sin(θ)
1. It’s periodic with period 2π and | sin(θ) | ≤ 1 for all θ.
sin(θ) =1
for θ = {π/2+2nπ)}
3. Sin(–θ) = –sin(θ) is odd so its graph is symmetric to
the origin.
Graphs of Trig–Functions
0 π 2π–π–2π 3π–3π
sin(θ) = –1
for θ = {–π/2+2nπ)}
sin(θ) = 0
for θ = {nπ} y = 1
y = –1
36. Graph of y = cos(θ)
The cosine function tracks the x-coordinates of
the position of the ant as it runs around the circle.
Graphs of Trig–Functions
θ
(1, 0)
cos(θ) = x
(x, y)
37. Graph of y = cos(θ)
The cosine function tracks the x-coordinates of
the position of the ant as it runs around the circle.
The coordinates of (x, y) of an angle θ, after rotating
90o counter-clockwise,
Graphs of Trig–Functions
θ
y
(1, 0)
cos(θ) = x
(x, y)
at θ
A rotation of π/2
gives the identity
cos(θ) = sin(θ + π/2)
θ
(x, y)
38. Graph of y = cos(θ)
The cosine function tracks the x-coordinates of
the position of the ant as it runs around the circle.
The coordinates of (x, y) of an angle θ, after rotating
90o counter-clockwise,
Graphs of Trig–Functions
θ
y
(1, 0)
cos(θ) = x
(x, y)
at θ
A rotation of π/2
gives the identity
cos(θ) = sin(θ + π/2)
π
θ
(x, y)
39. Graph of y = cos(θ)
The cosine function tracks the x-coordinates of
the position of the ant as it runs around the circle.
The coordinates of (x, y) of an angle θ, after rotating
90o counter-clockwise, become (–y, x), which is the
position corresponding to the of the angle (θ + π/2).
Graphs of Trig–Functions
θ
y
(1, 0)
cos(θ) = x
(x, y)
at θ
(–y, x) at
θ+π/2
A rotation of π/2
gives the identity
cos(θ) = sin(θ + π/2)
π
θ
(x, y)
40. Graph of y = cos(θ)
The cosine function tracks the x-coordinates of
the position of the ant as it runs around the circle.
The coordinates of (x, y) of an angle θ, after rotating
90o counter-clockwise, become (–y, x), which is the
position corresponding to the the angle (θ + π/2).
So cos(θ) = sin(θ + π/2),
Graphs of Trig–Functions
θ
y
(1, 0)
cos(θ) = x
(x, y)
at θ
(–y, x) at
θ+π/2
A rotation of π/2
gives the identity
cos(θ) = sin(θ + π/2)
π
θ
(x, y)
41. Graph of y = cos(θ)
The cosine function tracks the x-coordinates of
the position of the ant as it runs around the circle.
The coordinates of (x, y) of an angle θ, after rotating
90o counter-clockwise, become (–y, x), which is the
position corresponding to the the angle (θ + π/2).
So cos(θ) = sin(θ + π/2), i.e. the graph of y = cos(θ)
is the sine graph shifted left by π/2.
Graphs of Trig–Functions
θ
y
(1, 0)
cos(θ) = x
(x, y)
at θ
(–y, x) at
θ+π/2
A rotation of π/2
gives the identity
cos(θ) = sin(θ + π/2)
π
θ
(x, y)
42. Graphs of Trig–Functions
A rotation of π/2 gives that cos(θ) = sin(θ + π/2).
This means the graph of y = cos(θ) is the π/2-left-shift
of y = sin(θ),
Graph of y = cos(θ)
y = sin(θ)(π/2, 1)
π
(0, 0)
43. Graphs of Trig–Functions
A rotation of π/2 gives that cos(θ) = sin(θ + π/2).
This means the graph of y = cos(θ) is the π/2-left-shift
of y = sin(θ), e.g. the point (π/2, 1) is shifted to (0, 1)
and (0, 0) is shifted to (–π/2, 1).
Graph of y = cos(θ)
y = sin(θ)(π/2, 1)
π
(0, 0)
44. Graphs of Trig–Functions
y = sin(θ)
A rotation of π/2 gives that cos(θ) = sin(θ + π/2).
This means the graph of y = cos(θ) is the π/2-left-shift
of y = sin(θ), e.g. the point (π/2, 1) is shifted to (0, 1)
and (0, 0) is shifted to (–π/2, 1).
y = cos(θ)
(π/2, 1)
(0, 1)
Here is the graph of y = cos(θ) after shifting sin(θ).
π
(0, 0)
(–π/2,0)
Graph of y = cos(θ)
(π/2,0)
46. Graph of y = tan(θ)
Graphs of Trig–Functions
x
y
(x , y)
y
xtan()
Given an angle , tan() =
is the length as shown here,
which is also the slope of the dial.(1,0)
47. Graph of y = tan(θ)
Graphs of Trig–Functions
x
y
(x , y)
y
xtan()
Given an angle , tan() =
is the length as shown here,
which is also the slope of the dial.(1,0)
x
y t
1
~~
48. Graph of y = tan(θ)
Graphs of Trig–Functions
x
y
(x , y)
y
xtan()
Given an angle , tan() =
is the length as shown here,
which is also the slope of the dial.(1,0)
Tan(θ) is defined between ±π/2
but not at ±π/2.
x
y t
1
~~
49. Graph of y = tan(θ)
Graphs of Trig–Functions
x
y
(x , y)
y
xtan()
Given an angle , tan() =
is the length as shown here,
which is also the slope of the dial.(1,0)
Tan(θ) is defined between ±π/2
but not at ±π/2. Specifically,
as θ →π/2– , tan(θ) → ∞θ →π/2–
tan(θ) → ∞
x
y t
1
~~
50. Graph of y = tan(θ)
Graphs of Trig–Functions
x
y
(x , y)
y
xtan()
Given an angle , tan() =
is the length as shown here,
which is also the slope of the dial.(1,0)
Tan(θ) is defined between ±π/2
but not at ±π/2. Specifically,
as θ →π/2– , tan(θ) → ∞
as θ → –π/2+ , tan(θ) → –∞
θ →π/2–
tan(θ) → ∞
θ → –π/2+
tan(θ) → –∞
x
y t
1
~~
51. Graph of y = tan(θ)
Graphs of Trig–Functions
x
y
(x , y)
y
xtan()
Given an angle , tan() =
is the length as shown here,
which is also the slope of the dial.(1,0)
Tan(θ) is defined between ±π/2
but not at ±π/2. Specifically,
as θ →π/2– , tan(θ) → ∞
as θ → –π/2+ , tan(θ) → –∞
Here are some tan(θ) values:
θ →π/2–
tan(θ) → ∞
θ → –π/2+
tan(θ) → –∞
x
y t
1
~~
π/60 π/4 π/3
0 1/3 1 3 ∞
π/2 –
θ
tan(θ)
0–π/2+
0
–π/6
–1/3
–π/4
–1
–π/3
–3–∞
tan(θ)
θ
52. Graph of y = tan(θ)
Graphs of Trig–Functions
π/60 π/4 π/3
0 1/3 1 3 ∞
π/2 –
θ
tan(θ)
0–π/2+
0
–π/6
–1/3
–π/4
–1
–π/3
–3–∞
tan(θ)
θ
x
y
(x , y)
tan()
(1,0)
y = tan(θ)
–π/2 π/2
0
θ
Plot these points to
obtain the graph of
y = tan(θ).
53. Graph of y = tan(θ)
Graphs of Trig–Functions
π/60 π/4 π/3
0 1/3 1 3 ∞
π/2 –
θ
tan(θ)
0–π/2+
0
–π/6
–1/3
–π/4
–1
–π/3
–3–∞
tan(θ)
θ
x
y
(x , y)
tan()
(1,0)
y = tan(θ)
–π/2 π/2
0
θ
Plot these points to
obtain the graph of
y = tan(θ).
54. Graph of y = tan(θ)
Graphs of Trig–Functions
π/60 π/4 π/3
0 1/3 1 3 ∞
π/2 –
θ
tan(θ)
0–π/2+
0
–π/6
–1/3
–π/4
–1
–π/3
–3–∞
tan(θ)
θ
x
y
(x , y)
tan()
(1,0)
y = tan(θ)
–π/2 π/2 3π/2–3π/2
0
The basic periodic interval
for tan(θ) is (–π/2, π/2) with
period π
θ
Plot these points to
obtain the graph of
y = tan(θ).
55. Graph of y = tan(θ)
Graphs of Trig–Functions
π/60 π/4 π/3
0 1/3 1 3 ∞
π/2 –
θ
tan(θ)
0–π/2+
0
–π/6
–1/3
–π/4
–1
–π/3
–3–∞
tan(θ)
θ
x
y
(x , y)
tan()
(1,0)
Plot these points to
obtain the graph of
y = tan(θ).
y = tan(θ)
–π/2 π/2 3π/2–3π/2
π–π 0
The basic periodic interval
for tan(θ) is (–π/2, π/2) with
period π and like sin(x),
tan(nπ) = 0
where n is an integer.
θ
56. Graph of y = cot(θ)
Graphs of Trig–Functions
x
y
(x , y)
y
x
cot()
Given an angle , cot() =
is the length as shown here.
(0, 1)
57. Graph of y = cot(θ)
Graphs of Trig–Functions
x
y
(x , y)
y
x
cot()
Given an angle , cot() =
is the length as shown here.
(0, 1)
Cot(θ) is defined between 0 and π
but not at 0 or π. Specifically,
as θ →0+ , cot(θ) → ∞
as θ → π– , cot(θ) → –∞
θ →0 +
cot(θ) →∞
θ → π–
cot(θ) → –∞
58. Graph of y = cot(θ)
Graphs of Trig–Functions
x
y
(x , y)
y
x
cot()
Given an angle , cot() =
is the length as shown here.
(0, 1)
Cot(θ) is defined between 0 and π
but not at 0 or π. Specifically,
as θ →0+ , cot(θ) → ∞
as θ → π– , cot(θ) → –∞
Here are some cot(θ) values:
θ →0 +
cot(θ) →∞
θ → π–
cot(θ) → –∞
π/6 0+π/4π/3
0 1/3 1 3 ∞
π/2
θ
cot(θ)
π
0
2π/3
–1/3
3π/4
–1
5π/6
–3–∞
cot(θ)
θ
π/2
59. Graph of y = cot(θ)
Graphs of Trig–Functions
x
y
(x , y)
cot()(0, 1)
π/6 0+π/4π/3
0 1/3 1 3 ∞
π/2
θ
cot(θ)
π
0
2π/3
–1/3
3π/4
–1
5π/6
–3–∞
cot(θ)
θ
π/2
60. Graph of y = cot(θ)
Graphs of Trig–Functions
x
y
(x , y)
cot()(0, 1)
π/6 0+π/4π/3
0 1/3 1 3 ∞
π/2
θ
cot(θ)
π
0
2π/3
–1/3
3π/4
–1
5π/6
–3–∞
cot(θ)
θ
π/2
We’ve the graph of y = cot(θ)
by plotting these points.
The periodic interval of cot(θ)
is (0, π),
y = cot(θ)
π/2
π–π 0
θ
2π
61. Graph of y = cot(θ)
Graphs of Trig–Functions
x
y
(x , y)
cot()(0, 1)
π/6 0+π/4π/3
0 1/3 1 3 ∞
π/2
θ
cot(θ)
π
0
2π/3
–1/3
3π/4
–1
5π/6
–3–∞
cot(θ)
θ
π/2
We’ve the graph of y = cot(θ)
by plotting these points.
The periodic interval of cot(θ)
is (0, π), and like cos(θ)
cot(θ) = 0 for
θ =
= {(2n+1)π/2} with n an integer.
–π
2 ,
π
2 ,
–3π
2 ,
3π
2 ,... .{ {
y = cot(θ)
–π/2 π/2 3π/2
π–π 0
θ
2π
62. Graph of y = cot(θ)
Graphs of Trig–Functions
x
y
(x , y)
cot()(0, 1)
π/6 0+π/4π/3
0 1/3 1 3 ∞
π/2
θ
cot(θ)
π
0
2π/3
–1/3
3π/4
–1
5π/6
–3–∞
cot(θ)
θ
π/2
We’ve the graph of y = cot(θ)
by plotting these points.
The periodic interval of cot(θ)
is (0, π), and like cos(θ)
cot(θ) = 0 for
θ =
= {(2n+1)π/2} with n an integer.
–π
2 ,
π
2 ,
–3π
2 ,
3π
2 ,... .{ {
y = cot(θ)
–π/2 π/2 3π/2
π–π 0
θ
2π
63. The reciprocal-ratios sec() = 1/cos() and
csc() = 1/sin(), the "secant" and the "cosecant" of ,
are used in science and engineering.
Graphs of Trig–Functions
64. The reciprocal-ratios sec() = 1/cos() and
csc() = 1/sin(), the "secant" and the "cosecant" of ,
are used in science and engineering.
Graphs of Trig–Functions
Their graphs may be obtained by "reciprocating"
the graphs of y = sin() and y = cos().
65. The reciprocal-ratios sec() = 1/cos() and
csc() = 1/sin(), the "secant" and the "cosecant" of ,
are used in science and engineering.
Graphs of Trig–Functions
Their graphs may be obtained by "reciprocating"
the graphs of y = sin() and y = cos().
Reciprocating Graphs
66. The reciprocal-ratios sec() = 1/cos() and
csc() = 1/sin(), the "secant" and the "cosecant" of ,
are used in science and engineering.
Graphs of Trig–Functions
Their graphs may be obtained by "reciprocating"
the graphs of y = sin() and y = cos().
Reciprocating Graphs
To form the reciprocal graph of a
continuous function, let's track the
position (x, 1/y) from (x, y).
y=1
67. The reciprocal-ratios sec() = 1/cos() and
csc() = 1/sin(), the "secant" and the "cosecant" of ,
are used in science and engineering.
Graphs of Trig–Functions
Their graphs may be obtained by "reciprocating"
the graphs of y = sin() and y = cos().
Reciprocating Graphs
To form the reciprocal graph of a
continuous function, let's track the
position (x, 1/y) from (x, y).
Points (x, y) where 0 < y < 1 are
reciprocated to the top with 1/y > 1,
and vice versa as shown.
y=1
(x, y)
(x, 1/y)
(x, 1/y)
(x, y)
(x, y)
(x, 1/y)
68. The reciprocal-ratios sec() = 1/cos() and
csc() = 1/sin(), the "secant" and the "cosecant" of ,
are used in science and engineering.
Graphs of Trig–Functions
Their graphs may be obtained by "reciprocating"
the graphs of y = sin() and y = cos().
Reciprocating Graphs
To form the reciprocal graph of a
continuous function, let's track the
position (x, 1/y) from (x, y).
Points (x, y) where 0 < y < 1 are
reciprocated to the top with 1/y > 1,
and vice versa as shown.
The points (x, 1)’s stay fixed
y=1(x, 1)
(x, y)
(x, 1/y)
(x, 1/y)
(x, y)
(x, y)
(x, 1/y)
69. The reciprocal-ratios sec() = 1/cos() and
csc() = 1/sin(), the "secant" and the "cosecant" of ,
are used in science and engineering.
Graphs of Trig–Functions
Their graphs may be obtained by "reciprocating"
the graphs of y = sin() and y = cos().
Reciprocating Graphs
To form the reciprocal graph of a
continuous function, let's track the
position (x, 1/y) from (x, y).
Points (x, y) where 0 < y < 1 are
reciprocated to the top with 1/y > 1,
and vice versa as shown.
The points (x, 1)’s stay fixed and
y=1
(x, y)
(x, 1/y)
(x, 1/y)
(x, y)
(x, 1)
(x, y)
(x, 1/y)
asymptotes are formed at (x,0)’s since 1/0 is UDF.
(x,0)
Vertical
Asymptotes
70. Graphs of Trig–Functions
Sine is periodic so we may reciprocate one period of
the sine wave to get the graph y = csc().
Reciprocating the y-coordinates
71. Graphs of Trig–Functions
Sine is periodic so we may reciprocate one period of
the sine wave to get the graph y = csc().
Reciprocating the y-coordinates π0 2π
( π/2, 1)
(3π/2, –1 )
y=sin()
y=csc()
72. Graphs of Trig–Functions
Sine is periodic so we may reciprocate one period of
the sine wave to get the graph y = csc().
Reciprocating the y-coordinates π0 2π
( π/2, 1)
(3π/2, –1 )
y=sin()
y=csc()
73. Graphs of Trig–Functions
Sine is periodic so we may reciprocate one period of
the sine wave to get the graph y = csc().
Reciprocating the y-coordinates π0 2π
( π/2, 1)
(3π/2, –1 )
y=sin()
y=csc()
The graph of csc() is periodic with period 2π.
74. Graphs of Trig–Functions
Sine is periodic so we may reciprocate one period of
the sine wave to get the graph y = csc().
Reciprocating the y-coordinates π0 2π
( π/2, 1)
(3π/2, –1 )
y=sin()
y=csc()
The graph of csc() is periodic with period 2π.
Since cos() is the left-shift of the sin(), so the graph
of sec() is the left shift of the graph of csc().
75. Graphs of Trig–Functions
Sine is periodic so we may reciprocate one period of
the sine wave to get the graph y = csc().
Reciprocating the y-coordinates π0 2π
( π/2, 1)
(3π/2, –1 )
y=sin()
y=csc()
The graph of csc() is periodic with period 2π.
Since cos() is the left-shift of the sin(), so the graph
of sec() is the left shift of the graph of csc().
Here are the graphs of all six trig-functions.
76. π/2 3π/20 2π
(1, π/2)
One shaded
period.
0 2π
Graphs of Trig–Functions
0
y = csc()
y = sec()
y = tan()
(0, 1) (2π, 1)
(2π, 1)
(3π/2, –1 )
(3π/2, –1 )