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

The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

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