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Graphing Sine, Cosine, Secant, Cosecant, Tangent, and Cotangent Functions, Lecture notes of Advanced Calculus

Instructions on how to graph basic sine and cosine functions, as well as their transformations, including secant, cosecant, tangent, and cotangent functions. It includes tables of values, key points, and formulas for transformations.

Typology: Lecture notes

2021/2022

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Section 5.2 - Graphs of the Sine and Cosine Functions
In this section, we will graph the basic sine function and the basic cosine function and then graph other sine
and cosine functions using transformations. Much of what we will do in graphing these problems will be the
same as earlier graphing using transformations.
Definition: A non-constant function f is said to be periodic if there is a number p > 0 such that
( ) ( )
f x p f x
+ =
for all
x in the domain of f. The smallest such number p is called the
period
of f.
The graphs of periodic functions display patterns that repeat themselves at regular intervals.
Definition:
For a periodic function f with maximum value M and minimum value m.
The
amplitude
of the function is:
.
2
mM
In other words the amplitude is half the height.
Example 1:
State the period and amplitude of the periodic function.
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pff
pf12
pf13
pf14
pf15
pf16
pf17
pf18
pf19
pf1a
pf1b
pf1c
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Section 5.2 - Graphs of the Sine and Cosine Functions

In this section, we will graph the basic sine function and the basic cosine function and then graph other sine and cosine functions using transformations. Much of what we will do in graphing these problems will be the same as earlier graphing using transformations.

Definition: A non-constant function f is said to be periodic if there is a number p > 0 such that f ( x + p ) = f ( ) x for all x in the domain of f. The smallest such number p is called the period of f.

The graphs of periodic functions display patterns that repeat themselves at regular intervals.

Definition: For a periodic function f with maximum value M and minimum value m.

The amplitude of the function is:. 2

Mm

In other words the amplitude is half the height.

Example 1: State the period and amplitude of the periodic function.

Example 2: State the period and amplitude of the periodic function:

Note: For a periodic function f , the period of the graph is the length of the interval needed to draw one

complete cycle of the graph. For a basic sine or cosine function, the period is 2 π.

For a basic sine or cosine function, the maximum value is 1 and the minimum value is -1, so the amplitude

is 1. 2

Drawing all of these points is rather tedious. We’ll ask you to learn the shape of the graph and just graph five basic points, the x and y intercepts and the maximum and the minimum.

Period: 2 π

Amplitude: 1

x-intercepts: π , 2π

y-intercept: (0,0)

Big picture: f ( x )=sin( x )

Period: 2 π

Amplitude: 1

x-intercepts: k π

y-intercept: (0,0)

Now we’ll repeat the process for the basic cosine function f ( x )= cos( x ). The domain of this function is

( − ∞, ∞)and the range is [-1, 1]. Again, we typically graph just one complete period of the graph, that is on

the interval [ 0 , 2 π].

Here is the table of values for f ( x )= cos( x ):

x 0

6

cos x

Now we’ll graph these ordered pairs.

Now we’ll turn our attention to transformations of the basic sine and cosine functions. These functions will be of the form f ( x )= A sin( BxC )+ D or g ( x )= A cos( BxC )+ D .We can stretch or shrink sine and

cosine functions, both vertically and horizontally. We can reflect them about the x axis, the y axis or both axes, and we can translate the graphs either vertically, horizontally or both. Next we’ll see how the values for A, B, C and D affect the graph of the sine or cosine function.

Graphing f^ ( ) x^^ =^ A^ sin(^ Bx^ −^ C^ )^ +^ D^ or^ g x ( )^^ =^ A^ cos(^ Bx^ −^ C^ )+ D

  • The amplitude of the graph of is A.
  • The period of the function is:.

B

  • If A > 1, this will stretch the graph vertically. 0 < A < 1, this will shrink the graph vertically If A < 0, the graph will be a reflection about the x axis.
  • If B >1 , this will shrink the graph horizontally by a factor of 1/B. If 0<B<1, this will stretch the graph horizontally by a factor of 1/B.
  • Vertical Shift: Shift the original graph D units UP if D > 0, D units DOWN if D < 0.
  • Phase shift: The function will be shifted B

C

units to the right if > 0 B

C

and to the left if < 0 B

C

. The

number B

C

is called the phase shift.

Note: Horizontal Shift: If the function is of the form f ( x )= sin( xC )or f ( x )= cos( xC ), then shift the original graph C units to the RIGHT if C > 0 and C units to the LEFT if C < 0.

Amplitude:

Transformations:

( ) cos 2

f x x

Period:

Amplitude:

Transformations:

f ( ) x = 2 cos ( x − π) Period:

Amplitude:

Transformations:

f ( ) x = −5sin 4 ( x + π) Period:

Amplitude:

Transformations:

f ( ) x = sin 2 ( x − π)+ 1 Period:

Amplitude:

Transformations:

( ) 5sin 1 2 8

x f x

 π^ π

Period:

Amplitude

Transformations:

Example 5: Sketch over one period: ( ) cos 2

x f x

  • Example 6: Sketch over one period: f ( ) x = 4 cos(2 π x ) −

Example 8: Sketch over one period:

( ) 2 cos 4 8

f x x

Exercise: Sketch over one period: ( ) 2sin 1 2 8

x f x

 π^ π

Exercise: Consider the graph: Write an equation of the form f ( x )= A sin( BxC )+ D and an equation of

the form f ( x )= A cos( BxC )+ D which could be used to represent the graph. Note: these answers are not

unique!

Section 5.3a - Graphs of Secant and Cosecant Functions

Using the identity , sin( )

csc( ) x

x = you can conclude that the graph of g will have a vertical

asymptote whenever sin( x ) = 0. This means that the graph of g will have vertical asymptotes at

x = 0 ,± π ,± 2 π, …. The easiest way to draw a graph of g ( x )= csc( x )is to draw the graph of

f ( x )= sin( x ), sketch asymptotes at each of the zeros of f ( x )= sin( x ), then sketch in the

cosecant graph.

1 ( ) csc( ) sin( )

g x x x

= = ; if sin( ) x = 0 , then g(x) has a vertical asymptote.

Here’s the graph of f ( x )= sin( x )on the interval (^)  

Next, we’ll include the asymptotes for the cosecant graph at each point where sin( x )= 0.