2.7 TAN, COT, SEC and CSC Graphs
HOMEWORK: 2.7: 21-24, 29-32, 35, 36, 39, 40
2.8: 19, 20, 22-25
y = tan x

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













Key Points: (-π/4, -1), (0,0), (π/4, 1)
Key asymptotes: x = π/2 + kπ
p. 1 of 19
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2.7 TAN, COT, SEC and CSC Graphs
y = cot x
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
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

Key points: (π/4, 1), (π/2, 0), (3π/4, -1)
Key Asymptotes: x = 0 + kπ
p. 2 of 19
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2.7 TAN, COT, SEC and CSC Graphs
y = sec x and y = csc x
To get these graphs, we will just use the identities:
sec x = 1/ cos x
and csc x = 1 / sin x
y = sec x
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
p. 3 of 19
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

2.7 TAN, COT, SEC and CSC Graphs

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







y= csc x
To graph y  Asec(x   ) or y  Acsc(x   )

p. 4 of 19



2.7 TAN, COT, SEC and CSC Graphs
Example: Graph y = 6 tan(2x – π/4) + 1 and clearly label all
important features. State the equations of the asymptotes and
the period.
Period _________
Phase shift ___________
Asymptotes in one cycle ____________________
General equation of asymptotes _________________
Range ____________
p. 5 of 19
2.7 TAN, COT, SEC and CSC Graphs
Example: Graph y = 3 cot(5x) and clearly label all important
features. State the equations of all asymptotes and the period.
Period _________
Phase shift ___________
Asymptotes in one cycle ____________________
General equation of asymptotes _________________
Range ____________
p. 6 of 19
2.7 TAN, COT, SEC and CSC Graphs
Example: Graph y = 2sec(3x – π/4). Clearly label the
coordinates of all key points and the asymptotes. State the
equations of the asymptotes and period.
Period _________
Phase shift ___________
Asymptotes in one cycle ____________________
General equation of asymptotes _________________
Range ____________
p. 7 of 19
2.7 TAN, COT, SEC and CSC Graphs
Example: Graph y = -4csc(2x – π/3). Clearly label the
coordinates of all key points and the asymptotes. State the
equations of the asymptotes and period.
Period _________
Phase shift ___________
Asymptotes in one cycle ____________________
General equation of asymptotes _________________
Range ____________
p. 8 of 19
2.7 TAN, COT, SEC and CSC Graphs
Example: Graph y =1 - 3sec(5x + π/2). Clearly label the
coordinates of all key points and the asymptotes. State the
equations of the asymptotes and period.
Period _________
Phase shift ___________
Asymptotes in one cycle ____________________
General equation of asymptotes _________________
Range ____________
p. 9 of 19
2.7 TAN, COT, SEC and CSC Graphs
Example: Given y = - csc(x/2 + π/3). State the equations of the
asymptotes and period in one cycle.
Period _________
Phase shift ___________
Asymptotes in one cycle ____________________
General equation of asymptotes _________________
Range ____________
p. 10 of 19
2.7 TAN, COT, SEC and CSC Graphs
Example: Graph y = - 5tan(3x + π/2). Clearly label the
coordinates of all key points and the asymptotes. State the
equations of the asymptotes and period.
Period _________
Phase shift ___________
Asymptotes in one cycle ____________________
General equation of asymptotes _________________
Range ____________
p. 11 of 19
2.7 TAN, COT, SEC and CSC Graphs
Example: Given y = 1-5cot(4x + π/2). State the equations of the
asymptotes and period.
Period _________
Phase shift ___________
Asymptotes in one cycle ____________________
General equation of asymptotes _________________
Range ____________
p. 12 of 19
2.7 TAN, COT, SEC and CSC Graphs
y
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
x
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y
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x
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y
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x
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y
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x
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The above graphs are of the equation
y  A tan(x)  B or y  Acot(x)  B and   0
Determine which equation and state if A and B are positive or
negative
p. 13 of 19
2.7 TAN, COT, SEC and CSC Graphs
y
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x
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
y
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x
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y
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x
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y
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x
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The above graphs are of the equation
y  A tan(x  )  B or y  Acot(x  )  B and   0 & A>0.
Determine which equation and if  is
0 + k*period or not 0 + k*period
p. 14 of 19

2.7 TAN, COT, SEC and CSC Graphs
Strategy for graphing trig functions
y  Atrig(x   )  B
Step 1: Identify A, B,  &  and base graph
{Note: Remember minus sign in equation!}
Step 2: Find 
Amp., period and phase shift h
Amp  A
(sine, cosine only)
** period 
2
if base sin, cos, sec, or csc.


** period  if base tan or cot


p.s. h 

Step 3: Find base points
3
,  1) ; (2 , 0)
2
2

3
Cosine : (0, 1); ( , 0) ; ( ,  1) ; ( , 0) ; (2 ,1)
2
2
Secant: Graph y  Acos(x   ) and draw asymptotes at zeros and

Sine: (0, 0); ( ,1) ; ( , 0) ; (
cups toward asymptotes. If B, shift vertically.
Cosecant: Graph y  Asin(x   ) and draw asymptotes at zeros and
cups toward
 asymptotes. If B, shift vertically




,), ( ,1), (0,0) , ( ,1), ( ,)
2
4
4
2



3
,1), ( ,)
Cotangent: (0,) , ( ,1), ( , 0) , (
4
2
4
Remember, if y value is, then there is an asymptote at x = the x




coordinate!
Tangent: (

p. 15 of 19




2.7 TAN, COT, SEC and CSC Graphs
Step 4: Transform the key points {Watch order!}
Horiz stretch:
x coord

or x 
1

Horiz shift: x coord  p.s.  x 
Vert stretch:
y coord  A


{NOT just  }
{NOT amp!}
Vert shift: y coord  B
Step 5: Plot resulting points and/or asymptotes and sketch graph.
Step 6: Draw additional cycle if needed. {Add new period to x values
or use scale}
Step 7: Verify amp, period, phase shift and asymptotes etc. are correct
for your graph. Check on your calculator.
p. 16 of 19
2.7 TAN, COT, SEC and CSC Graphs
Strategy to find asymptotes
Method 1: Transform the original asymptotes of the function.


For tangent, see where ( ,) and ( ,) went
2
2
For cotangent, see where (0,) and ( ,) went
For secant, findthe zeros of y Acos(x   )
 the zerosof y  Asin(x   )
For cosecant, find
find the primary asymptotes in one cycle and
For general asymptotes,
add k* new period

Method 2: Set x   = to old asymp. of base function + k* period.
y  Asec(x   )  B ,

3

x




2k


x



 2k and solve for x.
set
and

2
2
y  Atan(x   )  B



x





k


x



 k and solve for x.
set
and

2
2
y  Acsc(x   )  B
set x    0  2k and x      2k and solve for x.


y  Acot(x   )  B
set x    0  k and x      k and solve for x.
 p. 17 of 19

2.7 TAN, COT, SEC and CSC Graphs
True or False. If the answer is false, correct it

2
T F The period of y  3tan(5x  ) 1 is
3
5
T F The phase shift of y  3tan(5x 



3
T F The amplitude of of y  2sec(5x 


) 1 is 

3

15
) 1 is 2
T F Two consecutive asymptotes of y  2 cot(5x 


2
 and
15
15

T F Two consecutive asymptotes of y  2 cot(5x 

x

15
and x 
2
15

T F Two consecutive asymptotes of y  2 tan(5x 

x

15
and x 
2
15


3

3

3
) 1 are
) 1 are
) 1 are
T F The general equation of the asymp. of y  2sec(5x 

are x 
p. 18 of 19

20
 2k and x 

4

 2k

4
) 1
2.7 TAN, COT, SEC and CSC Graphs
T F The general equation of the asymptotes of y  2sec(5x 
are x 

2k
 2k
and x  
20
5
4
5
T F The general equation of the asymptotes of y  2 tan(5x 

are x  
4
 2k
3 2k

and x 

20
5
20
5

4

 0, then the graph of y  A tan(x  )  B
T F If   0 & A<
y



x


 










could look like

T F If   0 & A< 0, then the graph of y  Acot(x   )  B
y




could look like

 

x









2
T F The period of y  3csc(5x  ) 1 is
3
5
p. 19 of 19
) 1







) 1