Transcript “Teach A Level Maths” Vol. 1: AS Core Modules 38: The graph of tanq © Christine Crisp.

“Teach A Level Maths”
Vol. 1: AS Core Modules
38: The graph of tanq
© Christine Crisp
The Graph of tanq
Module C2
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The Graph of tanq
We are going to sketch the graph of tan q where
q is an angle between 0  and 90  .
The Graph of tanq
y
Draw a circle, radius 1,
with centre at the origin
and complete a triangle.
P (x, y)
1
From the triangle,
sinq  y and cosq  x
y
Also, tan q 
x
sinq
So, tan q 
cos q
y
x
q
O
x
N
To sketch the graph of tan q we can divide values
of sinq and cos q taken from their graphs.
The Graph of tanq
The graphs of sinq and cos q for 0   q  360  are
y  sinq
y
q
y  tanq
x
y
q
y  cosq
sin 0
0
0

cos 0
1
The Graph of tanq
The graphs of sinq and cos q for 0   q  360  are
y  sinq
y
q
y  tanq
x
x
y
q
y  cosq
sin 180
0
0

cos 180
1
The Graph of tanq
The graphs of sinq and cos q for 0   q  360  are
y
y  sinq
q
This line, where tanq is
not defined is called an
asymptote.
y  tanq
x
x
y
q
y  cosby
q zero gives infinity so
Dividing
tanq is not defined when q  90.
sin 90
1 

cos 90
0
x
The Graph of tanq
The graphs of sinq and cos q for 0   q  360  are
y  sinq
y
q
y  tanq
x
y
q
y  cosq
x
x
x
x
x
x
The Graph of tanq
The graphs of sinq and cos q for 0   q  360  are
y  sinq
y
q
x
x
y
y  tanq
x
x
x
q
y  cosq
x
x
x
x
x
x
The Graph of tanq
The graphs of sinq and cos q for 0   q  360  are
y  sinq
y
q
x
x
y
y  tanq
x
x
x
q
y  cosq
x
x
x
x
x
x
The Graph of tanq
y  tanq
q
The graph of y  tanq repeats every 180. . .
The Graph of tanq
tan q is defined for angles less than 0  and
greater than 360  in the same way as the other
trig functions so the graph can be extended.
e.g.
y  tanq
q
. . . it is cyclic with a period of 180 .
The Graph of tanq
Exercise
Sketch the graph of y  tanq for values of q
from  180 to 180 clearly showing the asymptotes.
Use the graph to give a value of q between  180
and 180 ( not equal to the given angle! ) where
(a) tan q  tan 30
(b) tan q  tan 160
(c) tan q   tan 70  (d) tan q  tan ( 40) 
(a)  150
Solution:
x
 150
x
30 
The Graph of tanq
Exercise
Sketch the graph of y  tanq for values of q
from  180 to 180 clearly showing the asymptotes.
Use the graph to give a value of q between  180
and 180 ( not equal to the given angle! ) where
(a) tan q  tan 30
(b) tan q  tan 160
(c) tan q   tan 70  (d) tan q  tan ( 40) 
(a)  150
 20
x
160
x
(b)  20
The Graph of tanq
Exercise
Sketch the graph of y  tanq for values of q
from  180 to 180 clearly showing the asymptotes.
Use the graph to give a value of q between  180
and 180 ( not equal to the given angle! ) where
(a) tan q  tan 30
(b) tan q  tan 160
(c) tan q   tan 70  (d) tan q  tan ( 40) 
(a)  150
x
 70
x
70 
110
(b)  20
(c)  70  or 110
x
The Graph of tanq
Exercise
Sketch the graph of y  tanq for values of q
from  180 to 180 clearly showing the asymptotes.
Use the graph to give a value of q between  180
and 180 ( not equal to the given angle! ) where
(a) tan q  tan 30
(b) tan q  tan 160
(c) tan q   tan 70  (d) tan q  tan ( 40) 
(a)  150
 40
x
140
x
(b)  20
(c)  70  or 110
(d) 140
The Graph of tanq
The Graph of tanq
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photocopied.
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as “Handouts” with up to 6 slides per sheet.
The Graph of tanq
y
Draw a circle, radius 1,
with centre at the origin
and complete a triangle.
P (x, y)
From the triangle,
sinq  y and cosq  x
y
Also, tan q 
x
sinq
So, tan q 
cos q
1
y
x
q
O
x
N
To sketch the graph of tan q we can divide values
of sinq and cos q taken from their graphs.
The Graph of tanq
y  tanq
q
The graph of y  tanq repeats every 180. . .
. . . it is cyclic with a period of 180 .