41: Trig Equations41: Trig Equations

© Christine Crisp

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

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

Trig Equations

Module C2

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Trig Equations

xy sin

BUT, by considering the graphs of and , we can see that there are many more solutions:

xy sin 50y

e.g.1 Solve the equation .

50sin x

Solution: The calculator gives us the solution x =

30

50y

Every point of intersection of and gives a solution ! In the interval shown there are 10 solutions, but in total there are an infinite number.

xy sin 50y

The calculator value is called the principal solution

30

principal solution

Trig Equations

0180 360

xy sin

1

-1

We will adapt the question to:

Solution: The first answer still comes from the calculator: 30x

50yAdd the line 50y

Solve the equation for

50sin x 3600 x

3600 xx andSketch between xy sin

There are 2 solutions.

The symmetry of the graph . . . 15030180 x

15030

. . . shows the 2nd solution is

It’s important to show the scale.

Tip: Check that the solution from the calculator looks

reasonable.

Trig Equations

0 180 360

xy cos

1

-1

Solution: The first answer from the calculator is 120x

50y

Add the line 50y

e.g. 2 Solve the equation in the interval

50cos x3600 x

3600 xx andSketch between xy cos

There are 2 solutions.

The symmetry of the graph . . . 240120360 x

240120

. . . shows the 2nd solution is

Trig Equations

0180 360

xy sin

0 180 360

xy cos

SUMMARY

• Find the principal solution from a calculator.

• Find the 2nd solution using symmetry

where c is a constant

To solve

cx sin 3600 xor cx cos for

or

• Draw the line y = c.

• Sketch one complete cycle of the trig function. For example sketch from to .

3600

Trig Equations

0 180 360

xy cos

50y1

-1

Exercises

30060

The 2nd solution is

60360 x300

1. Solve the equations (a) and (b) for50cos x 3600 x

23sin x

Solution: (a) ( from calculator )60x

Trig Equations

0180 360

xy sin

1

-1

Solution: ( from calculator )

60x

23y

12060

The 2nd solution is

60180 x120

(b) ,sin23x 3600 x

Exercises

Trig Equations

y

90

2

-2

180 27090x

360

More Examplese.g. 3 Solve the equation in the

interval

giving answers correct to

the nearest whole degree.

2tan x 3600 x

Solution: ( from the calculator )63x

2y

24363

The 2nd solution is

24363180 x

Trig Equations

y

90

2

-2

x360180 27090

2y

24363

So all solutions to the equation can be found by repeatedly adding or subtracting to the first value.

1802tan x

Notice that the period of is and there is only one solution to the equation in each interval of .

xy tan 1802tan x

180

Trig Equations

So, to solve for 2tan x 720180 x

18063x 117

24318063 x

423180243 x 603180423 x

63xPrincipal solution:

This process is easy to remember, so to solve there is no need to draw a graph.

cx tan

First subtract

180

Now add to180 63

and keep adding . . .

Ans:

603,423,243,63,117x

Trig Equations

Solve the equation for50tan x 360180 x

Exercise

Solution: Principal

value

27x

15318027 x 333180153 x

Ans: 333,153,27

Adding 180

Trig Equations

xy sin

e.g. 4 Solve the equation forgiving the answers correct to 2 d. p.

70sin x x

780 x

780x( Because of the interval, it’s convenient to sketch from to . )

Switching the calculator to radians, we get

Solution:

implies radians

780372

2nd solution:

372x

Ans:

372,780

70y

More Examples

Trig Equations

Solution: ( from the calculator )30x

e.g. 5 Solve the equation for50sin x 3600 x

This value is outside the required interval . . . . . . but we still use it to solve the equation.

Tip: Bracket a value if it is outside the interval.

We extend the graph to the left to show 30x

More Examples

Trig Equations

xy sin

y

180

1

-1

x360

180

e.g. 5 Solve the equation for50sin x 3600 x

50y

30330

Since the period of the graph is this solution . . .

360

33036030 . . . is

Solution:

)30( x

More Examples

Trig Equations

xy sin

y

180

1

-1

x360

180

Solution:

e.g. 5 Solve the equation for5.0sin x 3600 x

50y

21030330

21030180

Symmetry gives the 2nd value for .3600 x

)30( x

The values in the interval are and 3302103600 x

More Examples

Trig Equations

So, if more solutions are required we add ( or subtract ) to those we already have.360

The graphs of and repeat every .360

xy sin xy cos

For solutions in the interval ,

720180 xwe also have

30360330 150360210

570360210 690360330

and 210x 330

e.g. In the previous example, we had )3600(

x

Trig Equations

0 180 360

xy cos

1

-1

66 294

Solution: Principal

value

66xe.g. 6 Solve for 40cos x 360180 x

29466360 xBy symmetry, 66360294 x

Method

140y

Ans: 294,66,66

Subtract from : 360 294

( is outside the interval ) 29436066 x

Trig Equations

1

-1

180180

xy cos

x

y

40y

6666

Solution: Principal

value

66xe.g. 6 Solve for 40cos x 360180 x

29436066 x

The solution can be found by using the symmetry of about the y-axis xy cos

66xMethod

2

Ans: 294,66,66Add to : 360 66

Trig Equations

SUMMARY

To solve or cx sin cx cos

360• Once 2 adjacent solutions have been

found, add or subtract to find any others in the required interval.

• Find the principal value from the calculator. • Sketch the graph of the trig function showing at least one complete cycle and including the principal value.

• Find a 2nd solution using the graph.

cx tan To solve• Find the principal value from the

calculator. • Add or subtract to find other solutions.

180

Trig Equations

1. Solve the equations ( giving answers

correct to the nearest whole degree ) 180180 x20sin x

(b) for650cos x 360180 x

(a) for

Exercises

Trig Equations

xy sin

y

180

1

-1

x

20y

360180

12

20sin x(a) for

168

Solution: Principal

value

12x

16812180 xBy symmetry,

Ans: 16812 ,

180180 x

Exercises

Trig Equations

180180

x

-1

y

1

xy cos

0 180 360

1

-1 xy cos

4949

31136049 x

49x

Ans: 311,49,49

(b) for650cos x 360180 x

Solution: Principal

value

49x

Either: Or:

31149360 x 49360311 x

311

650y 650y

49

Exercises

Trig Equations

Trig Equations

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.

Trig Equations

xy sin

Solution: The first answer comes from the calculator: 30x

50yAdd the line 50y

Solve the equation for

50sin x 3600 x

3600 xx andSketch between xy sin

There are 2 solutions.

The symmetry of the graph . . . 15030180 x

15030

. . . shows the 2nd solution is

e.g. 1

Trig Equations

xy cos

Solution: The first answer from the calculator is 120x

50y

Add the line 50y

e.g. 2 Solve the equation in the interval

50cos x3600 x

3600 xx andSketch between xy cos

There are 2 solutions.

The symmetry of the graph . . . 240120360 x

240120

. . . shows the 2nd solution is

Trig Equations

e.g. 3 Solve for 2tan x 720180 x

18063x 117

24318063 x

423180243 x 603180423 x

63xPrincipal solution:

This process is easy to remember, so to solve there is no need to draw a graph.

cx tan

First subtract

180

Now add to180 63

and keep adding . . .

Ans:

603,423,243,63,117x

Trig Equations

xy sin

e.g. 4 Solve the equation forgiving the answers correct to 2 d. p.

70sin x x

780 x

780x( Because of the interval, it’s convenient to sketch from to . )

Switching the calculator to radians, we get

Solution:

radians

780372

2nd solution:

372x

Ans:

372,780

70y

Trig Equations

Solution: ( from the calculator )30x

e.g. 5 Solve the equation for50sin x 3600 x

This value is outside the required interval . . .. . . but we still use it to solve the equation.

Tip: Bracket a value if it is outside the interval.We extend the graph to the left to show 30x

Trig Equations

xy sin

50y

21030 330

21030180

Symmetry gives the 2nd value as

Ans: , 330210

Since the period of the graph is , the 1st solution in is

360

33036030

3600 x

Trig Equations

66 294

Solution: Principal

value

66xe.g. 6 Solve for 40cos x 360180 x

29466360 xBy symmetry, 66360294 x

Method

140y

Ans: 294,66,66

Subtract from : 360 294

( is outside the interval ) 29436066 x

xy cos

Trig Equations

xy cos

40y

6666

29436066 x

The solution can be found by using the symmetry of about the y-axis xy cos

66x

Method

2

Ans: 294,66,66

Add to : 360 66

Trig Equations

SUMMARY To solve or cx sin cx cos

360• Once 2 adjacent solutions have been

found, add or subtract to find any others in the required interval.

• Find the principal value from the calculator. • Sketch the graph of the trig function showing at least one complete cycle and including the principal value.

• Find a 2nd solution using the graph.

cx tan To solve• Find the principal value from the

calculator. • Add or subtract to find other solutions.

180