38: Implicit Differentiation © Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core Modules.
41: Trig Equations © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.

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Transcript of 41: Trig Equations © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.
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 yaxis 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
192
Solution: Principal
value
12x
19212180 xBy symmetry,
Ans: 192,12
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
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 yaxis 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