Harder trig equations

42: Harder Trig 42: Harder Trig EquationsEquations

© Christine Crisp

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

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

Harder Trig Equations

"Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"

Module C2


360360

e.g.1 Solve the equation for the interval 180180

502sin

x

30x50sin x 1st solution:

180180

Sketch to find the 2nd solution:

Solution: Let so,2x 50sin x

( Once we have 2 adjacent solutions we can add or subtract to get the others. )360

There will be 4 solutions ( 2 for each cycle ).

We can already solve this equation BUT the interval for x is not the same as for .


0180 360

xy sin

50y

30 150 150,302 x

So,

33036030210360150 and

360360 xFor , the other solutions are

So, 150,30,210,3302 x

75,15,105,165 N.B. We must get all the solutions for x before we find . Alternate solutions for are NOT apart. 360

50sin x 360360 xfor


e.g. (a) forc4tan 1800 4x 7200 xUse and

We can use the same method for any function of .

c2

cose.g. (b) for 360360

180180 x2

xUse and

33030 x30xUse and

c )30sin( e.g. (c) for 3600


SUMMARY

Replace the function of by x.

Solving Harder Trig Equations

Write down the interval for solutions for x.

Find all the solutions for x in the required interval.

Convert the answers to values of .


0 180 360

xy cos

1

-1

50y

Exercise

30060

3600 7200 x60x50cos x

360300,36060,300,602 xSo,

330,210,150,30

1. Solve the equation for502cos 3600

Solution: Let 2x 50cos x

Principal value:

660,420,300,602 x


e.g.2

1cos x

If an exact value is not required, then switch the calculator to radian mode and get (3 d.p.)

c7850x

We sometimes need to give answers in radians. If so, we may be asked for exact fractions of .

Principal value is 4

Tip: If you don’t remember the fractions of ,use your calculator in degrees and then convert to radians using

radians

180

So, from the calculator452

1cos xx

4

x rads.


e.g. 2 Solve the equation giving exact answers in the interval .

013tan

The use of always indicates radians.Solution:

Let 3x

0 30 x

4

x ( or )

445

x

12

9,

12

5,

12

3

4

1st solution is

1tan x

For “tan” equations we usually keep adding to find more solutions, but working in radians we must remember to add .

180

4

9,

4

5,

43

x

42,

4,

43

xSo,


Solution: Let4

x 2

1cos x

e.g. 3 Solve the equation for the

interval . 2

1

4cos

20

45xPrincipal value:

2

1cos x

4

rads.

204

24

x

44

9 x

Sketch for a 2nd value:


0 2

xy cos

1

-1

702

1y

4

2nd value:

4

74

2 x

4

7x

repeats every , so we add to the principal value to find the 3rd solution:

2 2xcos

44

9 x

2

1cos x for

2,2

,03

Ans:

4

92

4

x

4

9,

4

7,

44

x

4

8,

4

6,0

4

x

2

2

3

1


e.g. 4 Solve the equation for

giving the answers correct to 2 decimal

places.

402

sin

x 40 x

We need to use radians but don’t need exact answers, so we switch the calculator to radian mode.

Solution: We can’t let so we use a capital X ( or any another letter ).

2

xx

Let so2

xX 40sin X

40 x2

40

X

Principal value:

)410( cX

Sketch for the 1st solution that is in the interval:

2

1


y1

-1

40y

X

Xy sin

4120 5533

2

1st solution is

c5533

2nd solution is

c41202

xX

c412022

xX c8725

8725

Multiply by 2: Ans:

20 X40sin X for

cc 7411,117 x ( 2 d.p.)

2

xX


1. Solve the equation for12tan 20

giving the answers as exact fractions of .

2. Solve the equation for250)60(sin 180180 giving answers correct to 1

decimal place.

Exercise


20 40 x

4

x1tan x

4

13,

4

9,

4

5,

42

x

8

13,

8

9,

8

5,

8

1. Solve the equation for12tan 20

Principal value:

Solution: Let 2x 1tan x

Add :

Solutions


180180 514x

250sin x

2. Solve the equation for250)60(sin 180180 giving answers correct to 1

decimal place.

120240 x

Principal value:

250sin x

Sketch for the 2nd solution:

Solutions

Solution: Let

60x


xy sin

y1

-1

x360180

)5165514180(,51460 x

514 )5165(

250y

The 2nd value is too large, so we subtract

360

250sin x 120240 xfor

5194360516560 x 574,5134 Ans

:

60Add :



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.


SUMMARY

Replace the function of by x.

Solving Harder Trig Equations

Write down the interval for solutions for x.

Find all the solutions for x in the required interval.

Convert the answers to values of .


360360

e.g. 1 Solve the equation for the interval 180180

502sin

x

30x50sin x 1st solution:

180180

Sketch to find the 2nd solution:

Solution: Let so,2x 50sin x

( Once we have 2 adjacent solutions we can add or subtract to get the others. )360

There will be 4 solutions ( 2 for each cycle ).

We can already solve this equation BUT the interval for x is not the same as for .


150,302 x

So,

33036030210360150 and

360360 xFor , the other solutions are

So, 150,30,210,3302 x

75,15,105,165 N.B. We must get all the solutions for x before we find . Alternate solutions for are NOT apart. 360

50sin x 360360 xfor

xy sin

50y

15030


e.g. (a) forc4tan 1800 4x 7200 xUse and

We can use the same method for any function of .

c2

cose.g. (b) for 360360

180180 x2

xUse and

33030 x30xUse and

c )30sin( e.g. (c) for 3600


The use of always indicates radians.

e.g. 2 Solve the equation giving exact answers in the interval .

013tan

Solution: Let

3x

0 30 x

4

x ( or )

445

x

4

9,

4

5,

43

x 12

9,

12

5,

12

3

4

1st solution is

1tan x

For “tan” equations we usually keep adding to find more solutions, but working in radians we must remember to add .

180


Solution: Let4

x 2

1cos x

e.g. 3 Solve the equation for the

interval . 2

1

4cos

20

45xPrincipal value:

2

1cos x

4

rads.

204

24

x

44

9 x

Sketch for a 2nd solution:


702

1y

4

2nd value:

4

7

4

7x

42

x

repeats every , so we add to the 1st value:

2 2xcos

44

9 x

2

1cos x for

2,2

,03

Ans:

4

92

4

x

4

9,

4

7,

44

x

4

8,

4

6,0

4

x

2

2

3

xy cos

So,


e.g. 4 Solve the equation for

giving the answers correct to 2 decimal

places.

402

sin

x 40 x

We need to use radians but don’t need exact answers, so we switch the calculator to radian mode.

Solution: We can’t let so we use a capital X ( or any another letter ).

2

xx

Let so2

xX 40sin X

40 x2

40

X

Principal value:

)410( cX

Sketch for 1st solution that is in the interval:

2

1


40y

X

Xy sin

4120 5533

1st solution is

c5533

2nd solution is

c41202

xX

c412022

xX c8725

8725

Multiply by 2: Ans:

20 X40sin X for

cc 7411,117 x ( 2 d.p.)

Education

Harder trig equations