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“Teach A Level Maths”
Vol. 1: AS Core Modules
42: Harder Trig Equations
Harder Trig Equations
e.g.1 Solve the equation sin 2 0 5 for the interval
0 360
Solution: Let x 2 so, sin x 0 5
There will be 4 solutions ( 2 for each cycle ).
1st solution: sin x 0 5 x 30
( Once we have 2 adjacent solutions we can add or
subtract 360 to get the others. )
Sketch to find the 2nd solution:
Harder Trig Equations
sin x 0 5
y 0 5
0
30
180
360
150
So,
x 2 30 , 150
y sin x
The other solutions are
360 30 390 and 360 150 510
x 2 30 , 150 ,390,510
So,
15 , 75 , 195, 255
N.B. We must get all the solutions for x before we find
. Alternate solutions for are NOT 360 apart.
Harder Trig Equations
SUMMARY
Solving Harder Trig Equations
Replace the function of
by x.
Convert the answers to values of
.
Harder Trig Equations
Exercise
1. Solve the equation cos 2 0 5 for 0 360
Solution: Let x 2
0 360
cos x 0 5
Principal value: cos x 0 5
x 60
1
y 0 5
0
-1
60
180
300
360
y cos x
So, x 2 60 , 300 , 60 360 , 300 360
x 2 60 , 300 , 420 , 660
30 , 150 , 210 , 330
Harder Trig Equations
e.g. 3 Solve the equation
cos 45
1
2
interval 0 360 .
Solution: Let
cos x
x 45
2
0 360
Principal value: cos x
1
2
Sketch for a 2nd value:
1
x 45
for the
Harder Trig Equations
1 for
45 x 405
cos x
2
y
1
2nd value:
07
2
0
-1
1
x 45
45
180
x 360 45
315 360
x 315
y cos x
cos x repeats every 360, so we add 360 to the
principal value to find the 3rd solution:
x 45 360 405
45 45, 315, 405
0, 270, 360
Harder Trig Equations
x
e.g. 4 Solve the equation sin 0 4 for 0 x 720
2
giving the answers correct to 2 decimal places.
x
Solution: We can’t let x
so we use a capital A
2
( or any another letter ).
Let
x
A
so sin A 0 4
2
0 x 720
Principal value: A sin 1 0.4 23.60
Sketch for the 1st solution that is in the interval:
Harder Trig Equations
sin A 0 4 for 0 A 360
y
x
A
2
1
23.6
203.6
336.4
180
180
X
360
y 0 4
-1
y sin X
x
A
180 23 .6 203.6
2
x
2nd solution is A
360 23 .6 336.4
2
Multiply by 2: Ans: x 7 11c , 11 74c ( 2 d.p.)
1st solution is
Harder Trig Equations
Exercise
1. Solve the equation sin ( 60 ) 0 25 for
180 180 giving answers correct to 1
decimal place.
Harder Trig Equations
Solutions
2. Solve the equation sin ( 60 ) 0 25 for
180 180 giving answers correct to 1
decimal place.
sin x 0 25
Solution: Let x 60
180 180
Principal value: sin x 0 25 x 14 5
Sketch for the 2nd solution:
Harder Trig Equations
sin x 0 25 for 240 x 120
y
1
y 0 25
180
14 5
-1
360
x
( 165 5 )
y sin x
x 60 14 5 , ( 180 14 5 165 5 )
The 2nd value is too large, so we subtract 360
x
60
165
5
360
194
5
Add 60 : Ans: 134 5 , 74 5
Harder Trig Equations
2sin(2x + 45°) = 1
Solution: Let y 2 x 45
0<x<360
2 sin y 1
0 x 360
2 sin y 1
sin y 0.5
Principal value: sin y 0 5 x 30
Harder Trig Equations
sin x 0 5
y
y 0 5
1
180
30
-1
360
x
150
y sin x
y 2 x 45 30 , (180 30 150 )
Add 360 to find further values : 390° , 510° , 750°
2 x 45 150 ,390 ,510 ,750
2x = 105°,345°,465°,705°
(subtract 45°)
x = 52.5°,172.5°,232.5°,352.5°
(divide by 2)
Harder Trig Equations
Harder 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.
Harder Trig Equations
SUMMARY
Solving Harder Trig Equations
Replace the function of
by x.
Write down the interval for solutions for x.
Find all the solutions for x in the required
interval.
Convert the answers to values of
.
Harder Trig Equations
e.g. 1 Solve the equation sin 2 0 5 for the interval
180 180
Solution: Let x 2 so, sin x 0 5
We can already solve this equation BUT the interval
for x is not the same as for .
180 180
360 x 360
There will be 4 solutions ( 2 for each cycle ).
1st solution: sin x 0 5 x 30
( Once we have 2 adjacent solutions we can add or
subtract 360 to get the others. )
Sketch to find the 2nd solution:
Harder Trig Equations
sin x 0 5 for 360 x 360
y 0 5
30
So,
x 2 30 , 150
150
y sin x
For 360 x 360, the other solutions are
150 360 210 and 30 360 330
So,
x 2 330 , 210 , 30 , 150
165 , 105 , 15 , 75
N.B. We must get all the solutions for x before we find
. Alternate solutions for are NOT 360 apart.
Harder Trig Equations
We can use the same method for any function of
e.g. (a) tan 4 c for
Use x 4
0 180
and 0 x 720
e.g. (b) cos c for 360 360
2
Use x
and 180 x 180
2
e.g. (c) sin( 30 ) c for
0 360
Use x 30 and 30 x 330
.
Harder Trig Equations
e.g. 2 Solve the equation tan 3 1 giving exact
answers in the interval 0 .
The use of
always indicates radians.
Solution: Let x 3
0
0 x 3
( or
st
)
x 45
tan x 1 1 solution is x
4
4
For “tan” equations we usually keep adding 180 to find
more solutions, but working in radians we must
remember to add .
5 9
5 3 9
x 3 ,
,
,
,
4 4
4
12 12 4 12
Harder Trig Equations
1
e.g. 3 Solve the equation cos
for the
4
2
interval 0 2 .
1
Solution: Let x
cos x
4
2
0 2
x 2
4
4
9
x
4
4
1
rads.
Principal value: cos x
x 45
4
2
Sketch for a 2nd solution:
Harder Trig Equations
1 for
9
cos x
x
4
4
2
y
1
x
4
07
2
So,
7
4
4
y cos x
7
x
value:
4
4
x repeats every 2 , so we add 2 to the 1st value:
2nd
cos
x 2
9
x 2
4
4
36 2 8
0,
,
24
4
7 9
x ,
,
4 4
4
4
Ans: 0,
3
2
, 2
Harder Trig Equations
x
e.g. 4 Solve the equation sin 0 4 for 0 x 4
2
giving the answers correct to 2 decimal places.
x
Solution: We can’t let x
so we use a capital X
2
( or any another letter ).
x
Let X
so sin X 0 4
2
2 4
0 x 4 0 X
21
We need to use radians but don’t need exact
answers, so we switch the calculator to radian mode.
Principal value: X ( 0 41c )
Sketch for 1st solution that is in the interval:
Harder Trig Equations
sin X 0 4 for 0 X 2
y sin X
0 412
3 553
5 872
X
y 0 4
x
1 solution is X 0 412c 3 553c
2
x
nd
2 solution is
X 2 0 412c 5 872c
2
Multiply by 2: Ans:
x 7 11c , 11 74c ( 2 d.p.)
st