38: The graph of 38: The graph of tantan

© Christine Crisp

““Teach A Level Maths”Teach A Level Maths”

Vol. 1: AS Core Vol. 1: AS Core ModulesModules

The Graph of tan

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Module C2

The Graph of tan

We are going to sketch the graph of where

tan

is an angle between and .

0 90

The Graph of tan

ysin

Draw a circle, radius 1, with centre at the origin and complete a triangle.

P (x, y)

x

y

O

y

1

Nx xcos

Also, x

ytan

So,

cos

sintan

To sketch the graph of we can divide values of and taken from their graphs.

tansin cos

From the triangle,and

The Graph of tan

tany

cosy

y

siny

y

00sin10cos

x

The graphs of and for are sin cos 3600

0

The Graph of tan

cosy

y

siny

y

x

The graphs of and for are sin cos 3600

x

0180sin1180cos 0

tany

The Graph of tan

siny

y

cosy

y

x

The graphs of and for are sin cos 3600

x x

190sin090cos

This line, where is not defined is called an asymptote.

tan

tany

Dividing by zero gives infinity so is not defined when .

tan 90

The Graph of tan

siny

y

cosy

y

x

The graphs of and for are sin cos 3600

x xx x

xx

tany

The Graph of tan

siny

y

cosy

y

x

The graphs of and for are sin cos 3600

x xx x

xx

x x

x x

tany

The Graph of tan

siny

y

cosy

y

x

The graphs of and for are sin cos 3600

x xx x

xx

x x

x x

tany

The Graph of tan

The graph of repeats every . . .tany 180

tany

The Graph of tan

is defined for angles less than and greater than in the same way as the other trig functions so the graph can be extended. e.g.

tan360

0

tany

. . . it is cyclic with a period of .180

The Graph of tan

Sketch the graph of for values of from to clearly showing the asymptotes.

tany180180

Exercise

(a)

30tantan

Use the graph to give a value of between and ( not equal to the given angle! ) where

(b)

160tantan (c

)

70tantan (d)

)40(tantan

Solution:

(a)

150

180180

x30

x150

The Graph of tan

Sketch the graph of for values of from to clearly showing the asymptotes.

tany180180

Exercise

(a)

30tantan

Use the graph to give a value of between and ( not equal to the given angle! ) where

(b)

160tantan (c

)

70tantan (d)

)40(tantan

(a)

150

180180

(b)

20x

160x

20

The Graph of tan

Sketch the graph of for values of from to clearly showing the asymptotes.

tany180180

Exercise

(a)

30tantan

Use the graph to give a value of between and ( not equal to the given angle! ) where

(b)

160tantan (c

)

70tantan (d)

)40(tantan

(a)

(b)

150

180180

20

x

70

(c)

70 110or x

110

x

70

The Graph of tan

Sketch the graph of for values of from to clearly showing the asymptotes.

tany180180

Exercise

(a)

30tantan

Use the graph to give a value of between and ( not equal to the given angle! ) where

(b)

160tantan (c

)

70tantan (d)

)40(tantan

(a)

(b)

(c)

150

180180

20

x

140

(c)

70 110or

(d)

140

x

40

The Graph of tan

The Graph of tan

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.

The Graph of tan

N

ysin

Draw a circle, radius 1, with centre at the origin and complete a triangle.

P (x, y)

x

y

O

y

1

x

xcos

Also, x

ytan

So,

cos

sintan

To sketch the graph of we can divide values of and taken from their graphs.

tansin cos

From the triangle,and

The Graph of tan

The graph of repeats every . . .tany 180

tany

. . . it is cyclic with a period of .180