Download Graphing Other Trigonometric Functions

Graphing Other Trigonometric
Functions
TRIGONOMETRY, 5.0 STUDENTS KNOW THE
DEFINITIONS OF THE TANGENT AND
COTANGENT FUNCTIONS AND CAN GRAPH
THEM.
Graphing Other Trigonometric Functions
Objective
Key Words
Graph tangent,
cotangent, secant,
and cosecant
functions.
2. Write equations of
trigonometric
functions
 Tangent
1.
 Cotangent
 Secant
 Cosecant
 Domain
 Range
 X-intercept
 Y-intercept
 Asymptote
Quick Check
 How many completely whole apples do you have if
you have 5/4 of an apple? So what is left?
 How many completely whole apples do you have if
you have ½ of an apple? So what is left?
 How many completely whole apples do you have if
you have 8 apples? So what is left?
 How would you express these three questions as an
algebraic expression? (Hint: apples, pieces of apples)
Quick Check
 Now think of π as the apple.
 How many completely whole π do you have if you have 5/4 of
an π? So what is left?
 How many completely whole π do you have if you have ½ of an
π? So what is left?
 How many completely whole π do you have if you have 8 π? So
what is left?

How would you express these three questions as an algebraic
expression? (Hint: π, pieces of π known as remainder)
Before We Begin, Recall the Unit Circle:
Trigonometric functions
 sin 𝜃 = 𝑦
 csc 𝜃 =
 cos 𝜃 = 𝑥
 tan 𝜃 =
𝑦
𝑥
=
sin 𝜃
cos 𝜃
 What are these x and y
values?
Reciprocal of
Trigonometric functions
 sec 𝜃 =
 cot 𝜃 =
1
𝑦
1
𝑥
𝑥
𝑦
 Take out your unit
circle to find out.
General Information you already know
The summary of Transformations for the sinusoidal
functions:
𝑦 = 𝐴 sin 𝐵 𝜃 − ℎ + 𝑘 and 𝑦 = 𝐴 cos 𝐵 𝜃 − ℎ + 𝑘
 The amplitude is 𝐴
 The period is
2𝜋
𝐵
 The horizontal shift is ℎ
 The midline is 𝑦 = 𝑘
 𝐵 is the angular frequency; that is, the number of
cycles completed in 0 ≤ 𝜃 ≤ 2𝜋
1: Graph Tangent
Example for Tangent of an Angle
Find each value by
referring to the graphs
of the trigonometric
functions.
tan 11π/4
Since 11π/4 = 2 + 3π/4,
Then tan 11π/4 = -1.
 You try: tan 7/2
1: Graph Cotangent
undefined
Example for Cotangent of an Angle
Find each value by
referring to the graphs
of the trigonometric
functions.
cot 11π/4
Since 5π/4 = 2 + π/2,
Then cot 5π/4 = 0.
 You try:
cot 3/2
1: Graph Cosecant
0
Example for Cosecant of an Angle
Find the values of  for
which each equation is
true.
csc  = -1
From the pattern of the
cosecant function, csc  =-1
if  = 3/2+ 2n, where n
is an integer.
 You try: csc θ = 1
1: Graph Secant
 = /2+ 2n
Example for Cosecant of an Angle
Find the values of  for
which each equation is
true.
sec  = -1
From the pattern of the
secant function, sec  = -1
if  = n, where n is an odd
integer.
 You try:
sec θ = 1
From the pattern of the secant function, sec  = 1 if  = n, where n is an even integer.
2: Graphing Trigonometric Functions
Order does matter!
y=A ???[B(θ-h)]+k
Draw the vertical shift,
k, and graph the
midline y=k. Use a
solid line.
2. Draw the amplitude,
𝐴 . Use dashed lines to
indicate the maximum
and minimum values of
the function.
3. Draw the period of
1.
the function,
2𝜋
,
𝐵
4. Graph the
appropriate
trigonometric curve.
5. Find the phase shift,
h, and translate the
graph accordingly.
2: Example for Graphing
Graph y=csc( - /2)+1.
 The vertical shift is 1. Use
this information to graph
the function.
 Amplitude is 1.
 The period is 2/1 or 2.
 The phase shift -(-/2/1)
or /2.
2: Example for Graphing
YOU TRY! Graph
y=csc(2 -/2)+1.
2: Example for Graphing
YOU TRY! Graph
y=csc(2 -/2)+1.
 The vertical shift is 1. Use
this information to graph
the function.
 The amplitude is 1
 The period is 2/2 or .
 The phase shift -(-/2/2)
or /4.
2: Example for Graphing
Write an equation for a
secant function with
period , phase shift
–π/2, and vertical shift
3.
 The vertical shift is k=3.



Substitute these values into
the general equation. The
equation is
y = sec (2 + ) + 3.

Thus, midline y=3
The amplitude is 1. Thus,
draw the dashed lines
above and below the
midline
The period π. Thus, B=2.
Draw the Secant curve
The phase shift is
h=-π/2
2: Example for Graphing
YOU TRY. Write an
equation for a secant
function with period ,
phase shift π/3, and
vertical shift -3.
 The vertical shift is k=-3.


Substitute these values into
the general equation. The
equation is
y = sec (2 -2/3)-3.


Thus, midline y=-3
The amplitude is 1. Thus,
draw the dashed lines
above and below the
midline
The period π. Thus, B=2.
Draw the Secant curve
The phase shift is
h=π/3
Conclusions
Summary
Assignment
 Remember the functions
 6.7: Graphing Other
tangent and cotangent
have a period of .
 Whereas sine and its
reciprocal function
cosecant and cosine and
its reciprocal function
secant both have periods
of 2.
Trigonometric
Functions

Pg400#(13-43 ALL, 45,48
EC)
 Problems not finished
are left as homework.