37: The graphs of37: The graphs ofsinsin and cos and cos
© Christine Crisp
““Teach A Level Maths”Teach A Level Maths”
Vol. 1: AS Core Vol. 1: AS Core ModulesModules
Graphs of
and
sin cos
Module C2
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Graphs of
and
sin cos
Trig Functions and Graphs
We are going to sketch the graph of where is an angle between and
. 0 90
sin
siny
Graphs of
and
sin cos
y sin
P (x, y)
x
y
O
Let P be a point with coordinates (x, y) on the circle with centre at the origin and radius 1.
Then, sin opp y
hyp 1
y
1
Let angle PON be
N
siny
Graphs of
and
sin cos
y1
20 40 60 80x
y
20y
e.g. 20 siny
siny
Graphs of
and
sin cos
y1
20 40 60 80x
y
40
y
e.g. 40
siny
Graphs of
and
sin cos
y1
20 40 60 80x
y
60
y
e.g. 60
siny
Graphs of
and
sin cos
y1
20 40 60 80
80
y
e.g. 80
x
y
siny
Graphs of
and
sin cos
y1
20 40 60 80x
y
90
y
e.g. 90
siny
Graphs of
and
sin cos
y1
20 40 60 80x
y
Also, when
,0
x
0y
x
x
x
x
x
siny
Graphs of
and
sin cos
x
x
x
x
xx
x
y
y1
20 40 60 80
siny
Graphs of
and
sin cos
As increases from to , y increases from 0 to 1.
0 90
y1
9020 40 60 8010 30 50 70
siny
siny
Graphs of
and
sin cos
We will now extend the definitions for an angle which can be any size and may be positive or negative.
We have, so far, only defined for angles between and .
sin900
siny
Graphs of
and
sin cos
yysin
x
y
30
y
30
Suppose , then
30 50sin
50
siny
Graphs of
and
sin cos
e.g. suppose , then 30 50sin
yysin 50
x
y
30
30
50
siny
Graphs of
and
sin cos
x
y
30
30150
y 150
50
If , . . .150
30150180 30
siny
Graphs of
and
sin cos
x
y
30
If , . . .150
y 150
. . . then y is still 50
ysinButSo,
50150sin
150
50
30
siny
Graphs of
and
sin cos
)180(sinsin
The quarter of the circle where the angles are between and is called the 1st quadrant. 900
For each angle in the 1st quadrant there is an angle of in the 2nd quadrant where
)180(
siny
Graphs of
and
sin cos
x
y
70
70110
110
e.g. 70 110)180( so
70sin110sin
Notice the symmetry about the y-axis .
)90(
siny
Graphs of
and
sin cos
sinyy
90
x x
70 110
e.g.
The symmetry about enables us to draw the graph for between and . 1800siny
90
siny
Graphs of
and
sin cos
The symmetry about enables us to draw the graph for between and .
1800siny
90
x x
70 110
e.g.
siny
y
90
siny
Graphs of
and
sin cos
The graph of between and is1800siny
siny
y
siny
Graphs of
and
sin cos
For angles between and ( the 3rd and 4th quadrants ), the y coordinates are negative.
180 360
30sin330sin
e.g.
330
210
210 330
x
y
3015050y50y
50y 50y
150sin210sin
So,
30sin
siny
Graphs of
and
sin cos
siny
y
The sine of each angle between and has the same magnitude as an angle between and BUT is negative. The graph for is:
3600
3600 180
180
siny
Graphs of
and
sin cos
For angles larger than we continue round the circle, so the values of repeat every .
360sin 360
e.g.
390
y
x 3050
360
siny
Graphs of
and
sin cos
We can extend the graph as shown
sinyy
siny
Graphs of
and
sin cos
siny
y
We can extend the graph as shown
siny
Graphs of
and
sin cos
Negative angles are defined as those formed by turning in a clockwise direction from .
0e.g. y
x 3050
30
30
30
So, 330sin)30(sin
Graphs of
and
sin cos
siny
y
If you have a graphical calculator, this graph will be one of the standard graphs
BUT make sure you can also sketch it without your calculator !
We can now draw the graph of for any interval. e.g.
siny
Graphs of
and
sin cos
SUMMARY
siny
y
• The trig function is defined for any angle.
sin
• The graph of repeats every .
siny 360
• The minimum value of is and the maximum is .
sin 11
• The graph for
is . . .
3600
. . . and must be memorised.
Graphs of
and
sin cosExercises
y
siny
1. Sketch the graph of for the intervalsiny
(a)
40sinsin
3600 Write down an angle between and ( not equal to the given angle! ) where
0 360
(b)
240sinsin (c
)
50sinsin (d)
280sinsin
(a)
140Ans:
x
40 140
x
Graphs of
and
sin cos
y
siny
1. Sketch the graph of for the intervalsiny
(a)
40sinsin
3600 Write down an angle between and ( not equal to the given angle! ) where
0 360
(b)
240sinsin (c
)
50sinsin (d)
280sinsin
(a)
140Ans:
x
240 300
x
(b)
300
Exercises
Graphs of
and
sin cos
y
siny
1. Sketch the graph of for the intervalsiny
(a)
40sinsin
3600 Write down an angle between and ( not equal to the given angle! ) where
0 360
(b)
240sinsin (c
)
50sinsin (d)
280sinsin
(a)
140(b
)
300(c
)
310230 or
Ans: x
50
230
x x
310
Exercises
Graphs of
and
sin cos
y
siny
1. Sketch the graph of for the intervalsiny
(a)
40sinsin
3600 Write down an angle between and ( not equal to the given angle! ) where
0 360
(b)
240sinsin (c
)
50sinsin (d)
280sinsin
(a)
140(b
)
300(c
)
310230 or (d
)
10080 or
Ans:
x x
80 100
x
280
Exercises
Graphs of
and
sin cos
Sketch a circle with radius 1 and centre at the origin O. Mark a point P (x, y) on the circumference in the 1st quadrant.
Using the sketch, write down an expression for .
cos
2(a) (b)
Mark , the angle between the radius OP and the x-axis.
(c)
(d)
(f) Refer to the circle to determine symmetry and hence complete the graph for .
Use the sketch to estimate enough values of to sketch the graph of for .
(e) 900 cos
cos
720360
Exercises
Graphs of
and
sin cos
2. Solution:
1
P (x, y)
x
y
O Nx
adjx
hyp(d) cos
Exercises
Graphs of
and
sin cos
2. Solution:
x
y
O
As increases, x decreases from 1 to 0
x = 1
xcos(d)
90
x = 0
3080x
6050x
e.g.
1,0 x80,30 x50,60 x
0,90 x
When we sketch the graph we use y instead of x.
Exercises
Graphs of
and
sin cos
cosyy
2. Solution:
Notice that is symmetric about( the y-axis ).
0cosy
Exercises
Graphs of
and
sin cos
Graphs of
and
sin cos
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Graphs of
and
sin cos
siny
The quarter of the circle where the angles are between and is called the 1st quadrant. 900
For each angle in the 1st quadrant there is an angle of in the 2nd quadrant where
)180(
)180(sinsin
Graphs of
and
sin cos
Notice that is symmetric about( the y-axis ).
0cosy
cosy
Graphs of
and
sin cos
• The trig functions and are defined for any angle.
sin
• The graphs repeat every .
360
• The minimum value is and the maximum is .
1 1
• The graphs for are3600
cos
siny cosy