32: The function 32: The function © Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core...

32: The function32: The function

© Christine Crisp

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

Vol. 2: A2 Core Vol. 2: A2 Core ModulesModules

xbxa sincos

The function xbxa sincos

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Module C3

Edexcel

Module C4

AQA

MEI/OCROCR


If you have either Autograph or a graphical calculator, draw the graph of

You will have the following graph:

xxy sin3cos4 On a calculator choose 360180 x 66 yand


xxy sin3cos4

It is clearly the shape of a sin or cos function but it has been transformed.

Can you describe, giving approximate values, the transformations from that give this curve?

xy cos

a stretch of s.f. 5 parallel to the y-axis, and

a translation of approx.

0

40

ANS:


xxy sin3cos4

The equation of the curve is approximately)40cos(5 xy

( As the cosine curve keeps repeating we could translate much further, for example but there’s no point doing this. )

)40036040(


We need an exact method of finding constants R and so that

Using the addition formula BABA sinsincoscos )cos( BA

the r.h.s. of (1) becomes )sinsincos(cos)cos( xxRxR

)cos(sin3cos4 xRxx )1(

Substitute in (1): )sinsincos(cossin3cos4 xxRxx

sinsincoscossin3cos4 xRxRxx

( R > 0 )


Since this is an identity, the l.h.s. and the r.h.s. must be exactly the same.

So, the term must be the same on both sides

xcos

sinxcos xsin xsinxcos cos4 3 R R





xcosand the term on both sides must be the same.

xsin



So, we can equate the coefficients



xcos

Coef. of :xcos

and the term on both sides must be the same.

xsin



cos4 R

Coef. of :xsin




xcos

Coef. of :xcos


xsin




cos4 R

Coef. of :xsin



xcos

Be careful to check the signs

Coef. of :xcos


xsin

sin3 R



)1(

)2(

We can now solve to find R and


cos4 R

Coef. of :xsin



xcos

Coef. of :xcos


xsin

sin3 R


Coef. of :xcos cos4 RCoef. of :xsin sin3 R

)1(

)2(

The easiest way to find is unusual.Divide (2) by (1):

cos

sin

4

3

R

R

4

3tan

This equation has an infinite number of solutions, but gives us the translation of the cosine curve. We can take the principal value which will translate the curve by the least amount.

936( 3 s.f. )

To find R we can square (1) and (2) and add them.

Why does this give R ?

xsin sinxcos xsin xcos cos4 3 R R



)1(

)2(

4

3tan 936 ( 3 s.f. )

222222 sincos34 RR

)sin(cos34 22222 R

:)2()1( 22

1sincos 22 AA But,222 34 R 22 34 R

R is positive because it gives the stretch from xy cos5R


)cos(sin3cos4 xxxSo, R



)1(

)2(

4

3tan 936 ( 3 s.f. )

222222 sincos34 RR

)sin(cos34 22222 R

:)2()1( 22

1sincos 22 AA But,222 34 R 22 34 R

5R)cos(sin3cos4 xxxSo, R



)936cos(5sin3cos4 xxx


)1(

)2(

4

3tan 936 ( 3 s.f. )

222222 sincos34 RR

)sin(cos34 22222 R

:)2()1( 22

1sincos 22 AA But,222 34 R 22 34 R

( 3 s.f. )5R

So,


The function xbxa sincos SUMMARY xx sin3cos4 To express in the form : )cos( xR

)cos( x• Expand using the addition formula sinsincoscos)cos( xxx

• Write sinsincoscossin3cos4 xRxRxx

• Equate the coefficients of xcos)1( xsin)2(and

• Divide the equations to find and solve for

tan• Square and add the equations

to get

222 34 R(or get this directly from the given

expression)• Choose the value of R > 0.


where a and b are constants and can be positive or negative.

We’ve used as an example of the more general function

xx sin3cos4

xbxa sincos

Suppose we have .xx sin12cos5 )cos(sin12cos5 xRxxLet

By choosing this addition formula, we have matched the signs on the l.h.s. and the r.h.s.cosBy choosing this addition formula, we have matched the signs on the l.h.s. and the r.h.s. So, and are both positive and is an acute angle. The translation from is less than

xcossin

90





xx sin3cos4

xbxa sincos

Suppose we have .xx sin12cos5 )cos(sin12cos5 xRxx


Can you complete this to find correct to 3 d.p. and R ?

Let


)cos(sin12cos5 xRxx


sin12 R)1(cos5 R

Coef. of :xsin

Coef. of :xcos

)2(sin12 R

)1(

)2(

cos

sin

5

12

R

R

5

12tan

467 13125 222 RR

)467cos(13sin12cos5 xxxSo,

Tip: If you have a graphical calculator, check by drawing both forms. They should give one curve.


To express as a single trig ratio, we’ve used

xbxa sincos

)sin( xR )sin( xRor

Since we can translate instead of to get curves of the same form, we can also use

xRsin xRcos

If you can choose which form to use, it’s better to choose the version which, when expanded, gives the same signs for the corresponding terms as the original expression.Just look in the formula book to see which of the 4 addition formulae match the expression.

)cos( xRand)cos( xR

The function xbxa sincos Exercis

e

)sin( xR

For each of questions (1) to (4), select the expression that would be easiest to use from the 4 below ,)cos( xR

xx cos4sin3 (1) For choose xx sin4cos3 (2) For choose

xx sin4cos3 (3) For choose

xx cos4sin3 (4) For choose

)sin( xR

)sin( xR

)cos( xR

)cos( xR

For (3) and (4) the terms could be switched so that either or could be used.

)cos( xR )sin( xR


One reason for expressing in one of the forms

xbxa sincos

)sin( xR)cos( xR or

e.g.)467cos(13sin12cos5 xxx

so the max. is 13 and the min is 13.

is that the stretch from or is obvious.

xcos xsin

The function xbxa sincos Exercis

eFor the following (i)express in the given form where R > 0

and , giving correct to 1 d.p.

)(xf900

(ii) write down the minimum and maximum value of

)(xf

xxxf sincos3)( 1. in the form )cos( xR

xxxf cos4sin3)( 2. in the form )sin( xR

3. in the form

xxxf cos4sin3)(

xxxf cos24sin7)( )sin( xR


Solution: )cos(sincos3 xRxx

sinsincoscossincos3 xRxRxx

sin1 R)1(cos3 R

Coef. of :xsin

Coef. of :xcos

)2(sin1 R

)1(

)2(

cos

sin

3

1

R

R

3

1tan

3021)3( 222 RR

)30cos(2sincos3 xxxSo,

The max. is 2 and the min is 2.

1. in the form )cos( xRxxxf sincos3)(


Solution: )sin(cos4sin3 xRxx

sincoscossincos4sin3 xRxRxx

sin4 R)1(cos3 R

Coef. of :xcos

Coef. of :xsin

)2(sin4 R

)1(

)2(

cos

sin

3

4

R

R

3

4tan

153543 222 RR

)153sin(5cos4sin3 xxxSo,


xxxf cos4sin3)( 2. in the form )sin( xRxxxf cos4sin3)(


Solution: )sin(cos24sin7 xRxx

sincoscossincos24sin7 xRxRxx )1(cos7 R

sin24 R )2(

)1(

)2(

cos

sin

7

24

R

R

7

24tan

77325247 222 RR

)773sin(25cos24sin7 xxxSo,


3. in the formxxxf cos24sin7)( )sin( xR

Coef. of :xcos

Coef. of :xsin


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 function xbxa sincos SUMMARY xx sin3cos4 To express in the form : )cos( xR

)cos( x• Expand using the addition formula sinsincoscos)cos( xxx

• Write sinsincoscossin3cos4 xRxRxx

• Equate the coefficients of xcos)1( xsin)2(and

• Divide the equations to find and solve for

tan• Square and add the equations

to get

222 34 R(or get this directly from the given

expression)• Choose the value of R > 0.



)1(

)2(

)sinsincos(cossin3cos4 xxRxx

4

3tan 936 ( 3 s.f. )

222222 sincos34 RR )sin(cos34 22222 R

:)2()1( 22

1sincos 22 AA But,222 34 R 22 34 R 5 R

)936cos(5sin3cos4 xxxSo, ( 3 s.f. )

e.g. Express in the formxx sin3cos4 )cos( xR

:)1(

)2(

Solution:

sinsincoscos)cos( xxx




xx sin3cos4

xbxa sincos

Suppose we have .xx sin12cos5

It is easier to solve for if and are positive so we choose to use

cos sin

)cos(sin12cos5 xRxx



sin12 R)1(cos5 R

Coef. of :xsin

Coef. of :xcos

)2(sin12 R

)1(

)2(

cos

sin

5

12

R

R

5

12tan

467 13125 222 RR

)467cos(13sin12cos5 xxxSo,

Tip: If you have a graphical calculator, check by drawing both forms. They should give one curve.



To express as a single trig ratio, we’ve used

xbxa sincos

)sin( xR )sin( xRor

Since we can translate instead of to get curves of the same form, we can also use

xRsin xRcos

If can choose which form to use, it’s better to choose the version which, when expanded, gives the same signs for the corresponding terms as the original expression.

Just look in the formula book to see which of the 4 addition formulae match the expression.

)cos( xRand)cos( xR


One reason for expressing in one of the forms

xbxa sincos

)sin( xR)cos( xR or

e.g.)467cos(13sin12cos5 xxx

so the max. is 13 and the min is 13.

is that the stretch from or is obvious.

xcos xsin

32: The function 32: The function © Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core...

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Transcript of 32: The function 32: The function © Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core...