42: Harder Trig 42: Harder Trig EquationsEquations
““Teach A Level Maths”Teach A Level Maths”
Vol. 1: AS Core Vol. 1: AS Core ModulesModules
Harder Trig Equations
e.g.1 Solve the equation for the interval 180180
502sin
30x50sin x 1st solution:
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 ).
Harder Trig Equations
0180 360
xy sin
50y
30 150 150,302 x
So,
33036030210360150 and
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
Harder Trig Equations
e.g. (a) forc4tan 1800 4xUse
We can use the same method for any function of .
c2
cose.g. (b) for 360360
2
xUse
30xUse
c )30sin( e.g. (c) for 3600
Harder Trig Equations
SUMMARY
Replace the function of by x.
Solving Harder Trig Equations
Convert the answers to values of .
Harder Trig Equations
0 180 360
xy cos
1
-1
50y
Exercise
30060
3600 60x50cos 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
Harder Trig Equations
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
)(45 0
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.
Harder Trig Equations
e.g. 2 Solve the equation giving exact answers in the interval .
013tan
The use of always indicates radians.Solution:
Let 3x
1800 45x
12
9
12
5
12
,,
1st solution is
1tan x
For “tan” equations we keep adding 180 to find more solutions.
405225453 ,, x
4536045180453 ,,xSo,
Work out the question using degrees and convert at the end
13575522 ,,.
Using Dr 180
Harder Trig Equations
Solution: Let 45x 2
1cos x
e.g. 3 Solve the equation for the
interval . 2
1
4cos
20
45xPrincipal value:
2
1cos x
3600
Sketch for a 2nd value:
Work out the question using degrees and convert at the end
Harder Trig Equations
0 180 360
xy cos
1
-1
702
1y
45
2nd value:
315
45360 x
315x
repeats every , so we add to the principal value to find the 3rd solution:
360 360xcos
40545 x2
1cos x for
2,2
,03
Ans:
40536045 x4053154545 ,,
45x
3602700 ,,
Using Dr 180
Harder Trig Equations
e.g. 4 Solve the equation for
giving the answers correct to 2 decimal
places.
402
sin
x 40 x
Solution: We can’t let so we use a capital A( or any another letter ).
2
xx
Let so2
xA 40 Asin
7200 x
Principal value:
01 62340 ..sin A
Sketch for the 1st solution that is in the interval:
Work out the question using degrees and convert at the end
Harder Trig Equations
y1
-1
40y
180X
Xy sin
623. 6203.
360180
1st solution is
6203.
2nd solution is
6231802
.x
A
6233602
.x
A 4336.
4336.
Multiply by 2: Ans:
3600 A40 Asin for
cc 7411,117 x ( 2 d.p.)
2
xX
Using Dr 180
Harder Trig Equations
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
Harder Trig Equations
3600
45x1tan x
585405225452 ,,, x
1. Solve the equation for12tan 20
Principal value:
Solution: Let 2x 1tan x
Add :180
Solutions
529252025112522 .,.,.,.
8
13,
8
9,
8
5,
8
Using Dr 180
Work out the question using degrees and convert at the end
Harder Trig Equations
180180 514x
250sin x
2. Solve the equation for250)60(sin 180180 giving answers correct to 1
decimal place.
Principal value:
250sin x
Sketch for the 2nd solution:
Solutions
Solution: Let
60x
Harder Trig Equations
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 :
Harder Trig Equations
2sin(2x + 45°) = 1 0<x<360
y2sin 1Solution: Let
2 45y x
y2sin 1 ysin 0.5
x0 360
x 30 Principal value:
ysin 0 5
Harder Trig Equations
xy sin
y
1
-1
x360180
y x2 45 30 , (180 30 150 )
30150
y 0 5
Add 360 to find further values : 390° , 510° , 750°
xsin 0 5
x 2 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
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 .
Harder Trig Equations
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 .
Harder Trig Equations
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
Harder Trig Equations
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
Harder Trig Equations
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
Harder Trig Equations
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:
Harder Trig Equations
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,
Harder Trig Equations
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
Harder Trig Equations
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.)