Transcript 41 Trig Equations

“Teach A Level Maths”
Vol. 1: AS Core Modules
41: Trig Equations
© Christine Crisp
Trig Equations
Module C2
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Trig Equations
e.g.1 Solve the equation sin x  0  5.
Solution: The calculator gives us the solution x = 30 
BUT, by considering the graphs of y  sin x and y  0 5,
we can see that there are many more solutions:
y  sin x
y  0 5
30 
principal solution
Every point of intersection of y  sin x and y  0 5
gives a solution ! In the interval shown there are 10
solutions, but in total there are an infinite number.
The calculator value is called the principal solution
Trig Equations
We will adapt the question to:
Solve the equation sin x  0  5 for 0  x  360
Solution: The first answer still comes from the
calculator:
x  30 
Sketch y  sin x between x  0 and x  360
Add the line
y  0 5
There are 2
solutions.
1
0
-1
y  0 5
30


150
180
360 
It’s important to show the scale.
y  sin x
Tip: Check that the solution from
the calculator
looks
The symmetry
of the graph
. .reasonable.
.
. . . shows the 2nd solution is x  180  30  150
Trig Equations
e.g. 2 Solve the equation cos x   0  5 in the
interval 0  x  360
Solution: The first answer from the calculator is
x  120
Sketch y  cos x between x  0 and x  360
Add the line
y   0 5
There are 2
solutions.
1
0
-1
120
180
240 
y  cos x
360 
y   0 5
The symmetry of the graph . . .
. . . shows the 2nd solution is x  360  120  240
Trig Equations
SUMMARY
 To solve sin x  c or cos x  c for 0  x  360
where c is a constant
•
•
Find the principal solution from a calculator.
Sketch one complete cycle of the trig
function. For example sketch from 0  to 360.
180
0
360
or
0
180
y  sin x
•
•
Draw the line y = c.
Find the 2nd solution using symmetry
y  cos x
360
Trig Equations
Exercises
1. Solve the equations
(a) cos x  0  5 and (b) sin x 
Solution: (a) x  60 
3
2
for 0  x  360
( from calculator )
1
y  0 5
0
60

180
300 
y  cos x
-1
The 2nd solution is
x  360  60
 300
360 
Trig Equations
Exercises
(b) sin x 
Solution:
3
2
,
x  60
0  x  360
( from calculator )
y  23
1
0
60 
120
180
y  sin x
-1
The 2nd solution is
360 
x  180  60
 120
Trig Equations
More Examples
e.g. 3 Solve the equation tan x  2 in the interval
0   x  360 giving answers correct to
the nearest whole degree.
Solution: x  63 ( from the calculator )
y
2
 90
y2
90 
63 
180
270 
243 
360 
x
-2
The 2nd solution is x  180  63  243
Trig Equations
y
2
 90
y2
90 
63 
180
270 
243 
360x
-2
Notice that the period of y  tan x is 180 and there
is only one solution to the equation tan x  2 in each
interval of 180.
So all solutions to the equation tan x  2 can be found
by repeatedly adding or subtracting 180 to the first
value.
Trig Equations
So, to solve tan x  2 for  180  x  720

Principal solution: x  63
First subtract 180
x  63  180   117
Now add 180 to 63 
x  63  180  243
and keep adding . . .
x  243  180  423
x  423  180  603
Ans: x  117 , 63 , 243 , 423 , 603
This process is easy to remember, so to solve
tan x  c there is no need to draw a graph.
Trig Equations
Exercise
Solve the equation tan x   0  5 for  180  x  360
Solution: Principal value
x   27 
Adding 180
x  27   180  153
x  153  180  333
Ans:  27  , 153 , 333
Trig Equations
More Examples
e.g. 4 Solve the equation sin x   0  7 for   x  
giving the answers correct to 2 d. p.
Solution:
implies radians
Switching the calculator to radians, we get
x   0 78
( Because of the interval, it’s convenient to sketch
from  to  . )
y  sin x
2 37
2nd solution:
x    0 78
 x  2 37
0 78
y  0 7
Ans:  0  78,  2  37
More Examples
Trig Equations
e.g. 5 Solve the equation sin x   0  5 for 0  x  360
Solution: x   30 ( from the calculator )
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 x   30
More Examples
Trig Equations
e.g. 5 Solve the equation sin x   0  5 for 0  x  360
Solution: x  (  30 )
y
1
 180
 30
180
y   0 5
-1
330  x
360 
y  sin x
Since the period of the graph is 360this solution . . .

. . . is
 30  360  330
More Examples
Trig Equations
e.g. 5 Solve the equation sin x   0.5 for 0  x  360
Solution: x  (  30 )
y
1
 180
210 
 30
180
y   0 5
-1
330  x
360 
y  sin x
Symmetry gives the 2nd value for 0  x  360
.


180  30  210
The values in the interval 0  x  360 are 210 and 330
Trig Equations
The graphs of y  sin x and y  cos x repeat
every 360 .
So, if more solutions are required we add ( or
subtract ) 360 to those we already have.
e.g. In the previous example, we had

 and



x  210
330
( 0 x 360 )
For solutions in the interval  180  x  720,
we also have
330  360   30
210  360   150
210  360  570
330  360  690
Trig Equations
e.g. 6 Solve cos x  0  4 for  180  x  360
Solution: Principal value x  66
Method 1
1
0
-1
y  0 4
66 
180
294
360 
y  cos x
By symmetry, x  360  66   294
Subtract 360 from 294: x  294  360   66

 is outside
( x  66  360
  294
Ans:
66 , 66 , 294 the interval )
Trig Equations
e.g. 6 Solve cos x  0  4 for  180  x  360
Solution: Principal value x  66
Method 2
The solution x   66  can be found by using the
symmetry of y  cos x about the y-axis
1
y  0 4
 180

 66
y
66 
-1
x
180
y  cos x
Add 360 to  66  : x   66  360  294
Ans:  66 , 66 , 294
Trig Equations
SUMMARY
 To solve sin x  c or cos x  c
• 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.
Once 2 adjacent solutions have been found, add
or subtract 360  to find any others in the
required interval.
 To solve tan x  c
• Find the principal value from the calculator.
• Add or subtract 180 to find other solutions.
Trig Equations
Exercises
1. Solve the equations ( giving answers correct to
the nearest whole degree )
for
 180  x  180
(b) cos x  0  65 for
 180  x  360
(a) sin x   0  2
Trig Equations
Exercises
(a) sin x   0  2 for  180  x  180
Solution: Principal value x  12
y
1
192
 12
 180
x
180
360
y   0 2
-1
By symmetry,
y  sin x
x  180  12  192
Ans:  12  , 192
Trig Equations
Exercises
(b) cos x  0  65 for  180  x  360
Solution: Principal value x  49
Either:
Or:
y
1
0
y  0 65
y  0 65
180
49 
311 360

1
 49
 180
y  cos x
-1

 360  49


 311

x
x  311  360   49

x
180
49 
-1
y  cos x
x   49
x   49  360  311
Ans:  49 , 49 , 311
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
e.g. 1 Solve the equation sin x  0  5 for 0  x  360
Solution: The first answer comes from the
calculator:
x  30 
Sketch y  sin x between x  0 and x  360
Add the line
y  0 5
y  0 5
There are 2
solutions.
30 
150
y  sin x
The symmetry of the graph . . .
. . . shows the 2nd solution is x  180  30  150
Trig Equations
e.g. 2 Solve the equation cos x   0  5 in the
interval 0  x  360
Solution: The first answer from the calculator is
x  120
Sketch y  cos x between x  0 and x  360
Add the line
y   0 5
There are 2
solutions.
120
y  cos x
240 
y   0 5
The symmetry of the graph . . .
. . . shows the 2nd solution is x  360  120  240
Trig Equations
e.g. 3 Solve tan x  2 for  180  x  720

Principal solution: x  63
First subtract 180
x  63  180   117
Now add 180 to 63 
x  63  180  243
and keep adding . . .
x  243  180  423
x  423  180  603
Ans: x  117 , 63 , 243 , 423 , 603
This process is easy to remember, so to solve
tan x  c there is no need to draw a graph.
Trig Equations
e.g. 4 Solve the equation sin x   0  7 for   x  
giving the answers correct to 2 d. p.
Solution:
radians
Switching the calculator to radians, we get
x   0 78
( Because of the interval, it’s convenient to sketch
from  to  . )
y  sin x
2 37
2nd solution:
x    0 78
 x  2 37
0 78
y  0 7
Ans:  0  78,  2  37
Trig Equations
e.g. 5 Solve the equation sin x   0  5 for 0  x  360
Solution: x   30 ( from the calculator )
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 x   30
Trig Equations
y  sin x
 30
210 
330 
y   0 5
Since the period of the graph is 360, the 1st
solution in 0  x  360 is

 30  360  330
Symmetry gives the 2nd value as


180  30  210
Ans: 210, 330
Trig Equations
e.g. 6 Solve cos x  0  4 for  180  x  360
Solution: Principal value x  66
Method 1
y  cos x
y  0 4
66 
294
By symmetry, x  360  66   294
Subtract 360 from 294: x  294  360   66
Ans:  66 , 66 , 294
( x  66  360   294 is outside the interval )
Trig Equations
Method 2
The solution x   66  can be found by using the
symmetry of y  cos x about the y-axis
y  0 4
 66
66 
y  cos x
Add 360 to  66  : x   66  360  294
Ans:  66 , 66 , 294
Trig Equations
SUMMARY
 To solve sin x  c or cos x  c
• 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.
Once 2 adjacent solutions have been found, add
or subtract 360  to find any others in the
required interval.
 To solve tan x  c
• Find the principal value from the calculator.
• Add or subtract 180 to find other solutions.