5.3 the graphs of sine and cosine...

5.3 The Graphs of Sine and Cosine Functions

Tim Busken

Grossmont College

Department of Mathematics

October 6, 2013

Sine function graph animation (Tim Fahlberg)Translating Trig Graphs Applet (Guillermo Bautista)

Tim Busken 5.3 The Graphs of Sine and Cosine Functions

http://math247.pbworks.com/w/page/20517676/Trigonometry%20with%20GeoGebra

http://gpbautista1.webs.com/geogebratutorial11/Trigonometric_Function.html

Consider an angle, x , located in standard position, such as the onegiven in the figure below. By definition, sin(x) is the secondcoordinate of the intersection of the terminal side of the angle withthe unit circle, and cos(x) is the first coordinate.

(1,0)

(0,-1)

(-1,0)

(0,1)(cos(x), sin(x))

xcos(x)

xsin(x)

1


f (x) = sin(x)

The graph of y = sin(x) has five key points between x = 0 andx = 2π, which I will refer to as quarter points.

x f (x)

0 0

π

21

π 0

3π2

−1

2π 0

y = sin(x)

−1

1

3π

2π

2π 2π

y

x

These five key points divide the x interval [0, 2π] into four equalparts. Notice the x coordinates of the five key points are the 90◦

(or quadrantal) type angles, and the y coordinates oscillatebetween the maximum at one and minimum at minus one.


x

1y = sin(x)

y

π 2π π3π

2

π

2

7π

2

5π

2 4π

−1

The domain of f (x) = sin(x) is all arc angles x , or real numbers x .

Therefore, the graph exists for x values outside of the interval[0, 2π]. Because sin(x + 2π) = sin(x) for every x , the exact shapeof the graph is repeated for x ∈ [2π, 4π], [4π, 6π], etc.;


y

x

1

−1

π 3π

22π−

3π

2−2π −π −

π

2

π

2

y = sin(x)

as well as for x ∈ [−2π, 0], x ∈ [−4π,−2π],... In fact the shaperepeats indefinitely over the set of real numbers. Furthermore,the range of y = sin(x) is y ∈ [−1, 1].


1

y

−1

(x, sin(x)) (x + p, sin(x + p))

x + px

Definition (Periodic Function)

If y = f (x) is a function and p is a nonzero constant such thatf (x) = f (x + p) for every x in the domain of f , then f is called aperiodic function. The smallest such positive constant p is theperiod of the function.


1

y

−1

(x, sin(x)) (x + p, sin(x + p))

x + px

The periodicity of the sine function is a result of the fact thatco-terminal angles have the same sine.


y

x

1

−1

π 2π−

3π

2−2π −π

y = sin(x)

3π

2−

π

2

Fundamental Cycle, 2π

π

2

The graph of y = sin(x) over any interval of 2π is called aone-period graph, the graph of one revolution, and/or onecycle of the sine wave.

The graph of y = sin(x) over [0, 2π] is called thefundamental cycle of y = sin(x).


−1

π 2πx

1

y

A

3π

2

QP

1 Pd = 2π

π

2

y = sin(x)

Definition (Amplitude)

The amplitude of the sine function is

Amplitude = |A| =|max −min|

2


−1

π 2πx

1

y

A

3π

2

QP

1 Pd = 2π

π

2

y = sin(x)

Definition (Quarter Point Width)

The Quarter Point Width (denoted QP) of the sine function is oneperiod length divided by four.

QP =Pd

4


f (x) = cos(x)

The graph of y = cos(x) also has five key points between 0 and 2π.

x f (x)

0 1

π

20

π −1

3π2

0

2π 1

x

y

y = cos(x)

3π

22πππ

2

1

−1

The five quarter points divide the x interval [0, 2π] into four equalquarter point widths (QP).


y

1

−1

x

y = cos(x)

−

3π

2−2π −π −

π

2π

2

π 3π

2

2π

Fundamental Cycle, 2π

1 The cosine function is periodic, which meanscos(x + p) = cos(x) for every x , therefore copies of the graph(and the five key points) for x ∈ [0, 2π] can be made toextend the graph over any domain.

2 The graph of y = cos(x) over [0, 2π] is called thefundamental cycle of y = cos(x).


−1

1

π 3π

2

x

y

π

22π

y = cos(x)

A

1 Pd = 2π

QP

Definition (Amplitude)

The amplitude of the cosine function is

Amplitude = |A| =|max −min|

2


−1

1

π 3π

2

x

y

π

22π

y = cos(x)

A

1 Pd = 2π

QP

Definition (Quarter Point Width)

The Quarter Point Width (denoted QP) of the cosine function isone period length divided by four.

QP =Pd

4


Both f (x) = sin(x) and f (x) = cos(x)

1 have domain x ∈ (−∞,∞),2 range y ∈ [−1, 1],3 fundamental period length, Pd = 2π,4 Amplitude, A = 1.

y

x

1

−1

π 3π

22π−

3π

2−2π −

π

2

π

2−π

y = sin(x)

y = cos(x)


Theorem

The amplitude of y = A · sin(x) or y = A · cos(x) is |A|.

−1

π 2π

1

y

3π

2

π

2

2

−2

y = 1.5 sin(x)

y = 2 sin(x)

y = sin(x)

x

−1

1

π 3π

2

x

y

π

22π

2

−2

y = 1.5 cos(x)

y = 2 cos(x)

y = cos(x)


If the value of A is negative, a reflection and amagnification is applied to the graph of y = sin(x) ory = cos(x).

−1

π 2π

1

3π

2

π

2

x

y

y = sin(x)

y = −1.5 sin(x)

−1.5

1.5

1

π 3π

2

x

y

π

22π

y = −1.5 cos(x)

y = cos(x)−1.5

−1

1.5

Theorem

The amplitude of y = A · sin(x) or y = A · cos(x) is |A|.


y

x

−1

π 2π−

3π

2−2π −π π

2−

π

23π

2

y = sin(x)

y = sin(x + 2)

2 units left

y

1

−1

x

−

3π

2−2π −π −

π

2π

2

π 3π

2

2π

y = cos(x)y = cos(x− 2)

2 units right

Definition (Phase Shift)

The phase shift, or horizontal shift of the graph of y = sin(x − C )or y = cos(x − C ) is the number C. The shift is to the right ifC > 0 and to the left if C < 0.

Example: For y = cos(x + π), is the shift to the right or left?


y = sin(x)

−1

π 2πx

1

y

3π

2

π

2

y = sin(2x)−1

1

π 3π

2

x

y

y = cos(x)

π

22π

y = cos(3x)

Definition (Changing the Period)

The period of the graph of y = sin(Bx) or

y = cos(Bx) is

Pd =2π

B.


y = sin(x)

−1

π 2πx

1

y

3π

2

π

2

y = sin(2x)−1

1

π 3π

2

x

y

y = cos(x)

π

22π

y = cos(3x)

Definition (Frequency)

The frequency, F , of the graph of y = sin(Bx) or

y = cos(Bx) is

F =1

Pd,

which represents the amount of time it takes to complete one cycleor revolution of the graph (if x represents the time axis).


−1

π 2πx

1

y

3π

2

π

2

y = sin(x)− 1

y = sin(x)

y = sin(x) + 1

−1

1

3π

2

x

y

π

22ππ

y = cos(x)

2

y = cos(x)− 1

y = cos(x) + 1

Theorem (Vertical Translation)

The graphs of y = sin(x) + D and y = cos(x) + D are vertical

translations of y = sin(x) and y = cos(x) a The vertical shift is

down if D < 0 and up if D > 0.


Definitions

The generalized sine and cosine families of functions can bedescribed by the two equations

f (x) = A sin[B (x −C )]+D and f (x) = A cos[B (x −C )] +D

where A,B ,C , and D are any real numbers.

|A| represents the amplitude, and

amplitude=|A| =|max −min|

2

The period is identified from Pd =2π

BC is the phase shift, or the horizontal shift.

D is the amount of vertical shift.

QP is the quarter-point width given by QP =Pd

4


5.3 the graphs of sine and cosine...

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