The graphs ofThe graphs ofsinsin and cos and cos

Trigonometric Ratios

TOA

CAH

SOHoppositex

hypotenusesin

oppositex

adjacenttan

adjacentx

hypotenusecos

S

O

H

C

A

H

T

O

A

Graphs of

and

s in cos

Trig Functions and Graphs

We are going to sketch the graph of where is an angle between and

. 0 90

s in

siny

Graphs of

and

s in cos

sin y

the height of the triangle

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

s in cos

y1

20 40 60 80x

y

20y

e.g. 20 siny

siny

Graphs of

and

s in cos

y1

20 40 60 80x

y

40

y

e.g. 40

siny

Graphs of

and

s in cos

y1

20 40 60 80x

y

60

y

e.g. 60

siny

Graphs of

and

s in cos

y1

20 40 60 80

80

y

e.g. 80

x

y

siny

Graphs of

and

s in cos

y1

20 40 60 80x

y

90

y

e.g. 90

siny

Graphs of

and

s in cos

y1

20 40 60 80x

y

Also, when

,0

x

0y

x

x

x

x

x

siny

Graphs of

and

s in cos

x

x

x

x

xx

x

y

y1

20 40 60 80

siny

Graphs of

and

s in 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

s in cos

s inyy

90

x x

70 110

e.g.

For angles between 90 and 180 the height of the triangle decreases

siny

Graphs of

and

s in cos

The symmetry about enables us to draw the graph for between and .

1800s iny

90

x x

70 110

e.g.

s iny

y

90

siny

Graphs of

and

s in cos

The graph of between and is1800siny

s iny

y

siny

Graphs of

and

s in cos

siny

y

For angles between and the height of the triangle is negative and so the graph appears under the horizontal axis

360180

siny

Graphs of

and

s in cos

We can extend the graph as shown

s inyy

siny

Graphs of

and

s in cos

s iny

y

We can extend the graph as shown

Graphs of

and

s in cos

s iny

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

s in cos

SUMMARY

siny

y

• The trig function is defined for any angle.

sin

• The graph of repeats every .

s iny 360

• The minimum value of is and the maximum is .

sin 11

• The graph for

is . . .

3600

. . . and must be memorised.

Graphs of

and

s in cosExercises

y

s iny

1. Sketch the graph of for the intervalsiny

(a)

40sins in

3600 Write down an angle between and ( not equal to the given angle! ) where

0 360

(b)

240s ins in

(a)

140Ans:

x

40 140

x

Graphs of

and

s in cos

y

s iny

1. Sketch the graph of for the intervalsiny

(a)

40sins in

3600 Write down an angle between and ( not equal to the given angle! ) where

0 360

(b)

240s ins in

(a)

140Ans:

x

240 300

x

(b)

300

Exercises

Graphs of

and

s in cos

y

s iny

1. Sketch the graph of for the intervalsiny

(a)

40sins in

3600 Write down an angle between and ( not equal to the given angle! ) where

0 360

(b)

240s ins in

(a)

140(b

)

300

Ans:

Exercises

Graphs of

and

s in cos

1

P (x, y)

x

y

O Nx

adjx

hyp cos

Graph of cos

cos x

the base of the triangle

Graphs of

and

s in cos

x

y

O

As increases, x decreases from 1 to 0

x = 1

cos x

90

x = 0

3080 x

6050 x

e.g.

1,0 x80,30 x50,60 x

0,90 x

When we sketch the graph we use y instead of x.

Graphs of

and

s in cos

oy=cosθy

Notice that is symmetric about( the y-axis ).

0cosy

Graphs of

and

s in cos

1. Sketch the graph of for the intervaly cos

(a)

cos cos 40

3600 Write down an angle between and ( not equal to the given angle! ) where

0 360

(b)

cos cos 240

(a)

320 Ans:

x

32040

x

(b)

oθ =300

Exercises

Graphs of

and

s in cos

1. Sketch the graph of for the intervaly cos

(a)

cos cos 40

3600 Write down an angle between and ( not equal to the given angle! ) where

0 360

(b)

cos cos 240

(a)

320 Ans:

(b)

120

Exercises

x

240120

x