Flexural and Lateral-Torsional Buckling Strengths of Double Angle ...
31 Double Angle Formulae
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Transcript of 31 Double Angle Formulae
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31: Double Angle31: Double Angle
FormulaeFormulae
Christine Crisp
Teach A Level MathsTeach A Level Maths
Vol. 2: A2 Core ModulesVol. 2: A2 Core Modules
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Double Angle Formulae
)5(BA
BA
tantan1
tantan
|)tan( BA
BABABAsincoscossin)sin(
|)1(
BABABA sinsinosos)os( | )3(
The double angle formulae are used to express an
angle such as 2A in terms of A.
We derive the formulae from 3 of the additionformulae.
What do we need to do to obtain formulae for
?tancos,sin AAA and
ANS: Replace B by A in(1),(3) and (5).
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Double Angle Formulae
BBB sinsincoscos)cos( |
BABABA sincoscossin)sin( |So,
AAAAA sincoscossin2sin |
AAAAA s ns ncoscos2cos |)2(AAA
22sincos2cos |
Using (2) in twoother ways:
,sincos22
| AA
)cos1(cos2cos22
AAA |
)1AAA cossin22sin |
)( a 1cc | AA
AAA22
s)s1(2cos |
)( b2
sin212cos
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Double Angle Formulae
)3(
B
B
tantan1
tantan
|)tan( B
Finally,
These formulae are probably NOT in your formulaebook but please check!
If they are not there, you need to remember them.
N.B. The formulae can be derived quite quickly fromthe addition formulae. However, you may notrecognise the need to use them unless you havememorised them.
AA
AAA
tantan1
tantan2tan
|
A
AA
2tan1
tan22tan
|
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Double Angle Formulae
SUMMARY
The double angle formulae are:
AAA cossin22sin | )1(
AAA22
sincos2cos | )2(
cos22
| A )( a
A2sin21 | )2( b
2tan1
tan22tan
| )3(
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Double Angle Formulae
N.B. The formulae link any angle with double the
angle.For example, they can be used for
x2 xand
x2
xand
y33 y
and
We use them to solve equations
to prove other identities
to integrate some functions
U4 and U2
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Double Angle Formulae
Ive then used the method I used in AS to find the
other solutions.
Some of you will use a different method and if you
are happy with it, dont change.Whichever method you use, make sure you get allthe required solutions!
In the following examples, once Ive reduced theequations to simple ones, Ive used a calculator tofind the principal values.
So, for the equations andI have sketched graphs.
c!sin c!cos
Solving Equations
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Double Angle Formulae
e.g. 1 Solve the following equation:
;0cossin ! xx Q3600 ee x
Solution:When solving trig equations we must aim for onlyone trig ratio of one angle.
Here we have two trig ratios ( cos and sin )
We can use AAA cinin | x2sin
0cos2sin ! xx
becomes 0coscossin2 ! xxx
We still have 2 trig ratios but is a commonfactor
xc
and two angles ( xand 2x).
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Double Angle Formulae
Solution:0coscossin2 ! xxx
0)1sin2(cos !xx
0cos !x or 01si !x
21sin !x
We now have 2 simple trig equations which we cansolve.
e.g. 1 Solve the following equation:
;0cossin ! xx Q3600 ee xDont cancel will give solutons. )
0cos !x
xcos
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Double Angle Formulae
xy cos!
!y
0cos !x Q
90!x ( principal value )
QQ270,90!x
Q90
Q270
e.g. 1 Solve the following equation:e.g. 1 Solve the following equation:
;0cossin ! xx Q3600 ee x
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Double Angle Formulae
xy sin!
Q30!x ( principal value )
QQ150,0!x
21!y
Q30
Q150
QQQQ270,150,90,30!xANS:
e.g. 1 Solve the following equation:
21si !x
e.g. 1 Solve the following equation:
;0cossin ! xx Q3600 ee x
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Double Angle Formulae
e.g. 2 Solve the following equation:
;3sin5cos ! xx T20 ee x
Solution:
Again we have 2 trig ratios and 2 angles.
AAA 22 sis2cos |
1cos|
A
A2
si21 |
By choosing the 3rd version . . .
We use the double angle formula for buhave 3 versions to choose from:
A2cos
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Double Angle Formulae
e.g. 2 Solve the following equation:
;3sin5cos ! xx T20 ee x
Solution:
Again we have 2 trig ratios and 2 angles.
AAA 22 sicos2cos |
1cos|
A
A2
si21 |
we will reduce the equation to 1 angle and 1 trig ratio.By choosing the 3rd version . . .
)2(
)2( a
)2( b
We use the double angle formula for buhave 3 versions to choose from:
A2cos
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Double Angle Formulae
e.g. 2 Solve the following equation:
;3sin5cos ! xx T20 ee x
Solution:
So, ;3s n52cos xxQuadraticequation3sin5sin2
2 xx
25202
ss
35212
ss )sin( xs !
)2)(12(0 ! ss
21i x 2si !xor
( no solutions )
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Double Angle Formulae
xy sin!
e.g. 2 Solve the following equation:
(b) ;3sin5cos ! xx T20 ee x
!21si x
21
y
67T
630(
TQx
( outside therequired range )
6
T
6
TTx ,
7T
611
62 TTT
6
11,
6
7 TTxANS:
6
11T
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Double Angle Formulae
e.g. 3 Solve the following equation giving the answer
correct to the nearest degree:;tan5tan UU ! QQ 180180 ee U
Solution: Use
A
AA
2
tan1
tan22tan
|
UU tan52tan2 !
U
U
Utan5
tan1
tan22
2!
tt
t5
1
42!
)1(54
2
ttt !
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Double Angle Formulae
;tan5tan UU ! QQ 180180 ee U
e.g. 3 Solve the following equation giving the answer
correct to the nearest degree:
)1(542ttt !
5
1
0 stt or
3554 ttt !
053 ! tt
0)5( 2 !tt
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Double Angle Formulae
;tan5tan UU ! QQ 180180 ee U
0tanU or
e.g. 3 Solve the following equation giving the answer
correct to the nearest degree:
447
5
1ta }
450 y
QQ180,0,180U
QQQ15618024
,24Q
!U
,24Q!UQQQ
15618024 !QQQQQQ
180156,4,0,4,156,180 !ANS:
Q24 Q156
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Double Angle Formulae
We can now use the double angle formulae to prove
other identities. The general method of proof isthe same as you have met before.
Proof:
U
UUU
sin
)1cos21(cossin22
|
l.h. . =
)coscos UU|
U3
cos4|
UU
UU 3
cos4si
)2cos1(2si
|
e.g. 1 Prove that
)0(sin {U
= r.h. .
( Double angle
formulae )
U
UU
sin
)2cos1(2sin
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Double Angle Formulae
Proof:
xx
xx
cossin
sincos22
|
e.g. 2 Prove thatx
xx
2sin
2tancot |
l.h. . =cos
sin
sin
cos
xxcossin
1|
xxcossin22|
x2sin
2| = r.h. .
)1sincos(22
| xx
( Multiplying by )2
2
( Double angleformula )
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Double Angle Formulae
Which double angle formula can we use to changethe function so that it can be integrated?
e.g. Find d2
si
ANS: AA2
sin212cos )2( b
Rearranging the formula:
AA2
si212cos |
)2cos1(si212 AA |
So, ! dxxdxx )2os1(sin 212
2
2sin x
!2
1x C
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Double Angle Formulae
The previous example is an important application of
a double angle formula.The next 2 are also important. Try them yourself.
Exercise
1. Find dxx2
cos
2. Find dcossin
( This one is a product. Why cant it beintegrated by parts ? )
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Double Angle FormulaeExercise
1. Find
dxx
2s
Solution: 1coscos | AA
AA2
21 co2co1 |
! dxxdxx )2cos1(cos 212So,
Cx
x
!
2
2sin
21
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Double Angle Formulae
2. Find dxxxcossin
AAA cossin22sin |Solution:
AAA cossin2sin2
1 |
! dxxdxxx 2sincossin 21
So,
Cx
!
2
2co
2
1
Cx
!4
cos
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Double Angle Formulae
SUMMARY
A2cos
The rearrangements of the double angle formulaefor are
)2cos1cos212
AA |
)2cos1(sin2
12 AA |
They are important in integration so you shouldeither memorise them or be able to obtain themvery quickly.