Use the formula
Area = 1/2bcsinA
Think about the yellow area
What’s sin (α+β)?
Sin(A+B) Ξ ?
Sin(A+B) Ξ ?
Sin(x+30) = ?
Sin(A+B) Ξ ?
Sin(x+30) = ?
Cos(A-B) Ξ ?
Sin(A+B) Ξ ?
Sin(x+30) = ?
Cos(A-B) Ξ ?
Cos(x-60) = ?
Sin(A+B) Ξ ?
Sin(x+30) = ?
Cos(A-B) Ξ ?
Cos(x-60) = ?
Tan(A+B) Ξ ?
Sin(A+B) Ξ ?
Sin(x+30) = ?
Cos(A-B) Ξ ?
Cos(x-60) = ?
Tan(A+B) Ξ ?
Tan(A+60) = ?
Use the formula
Area = 1/2bcsinA
Think about the yellow area
What’s sin (α+β)?
Trig addition Trig addition formulaeformulae
Aims: To learn the trig addition formula.
To solve equations and prove trigonometrical identities using
the addition formulae.
Trig Addition Formulae
1
Does ? 60sin30sin)6030sin(
and
371
So, 60sin30sin)6030sin(
We cannot simplify the brackets as we do in algebra because they don’t mean multiply.
90sin)6030sin(l.h.s. =
2
3
2
160sin30sin
r.h.s. =
Trig Addition Formulae
BA
BABA
tantan1
tantan)tan(
BABABA sincoscossin)sin(
BABABA sinsincoscos)cos(
The addition formulae are in your formulae booklets, but they are written as:
Notice that the cos formulae have opposite signs on the 2 sides.
Use both top signs in a formula or both bottom signs.
Trig Addition Formulae
Using the Addition Formulae e.g. Prove the following:
xyyxyx cossin2)sin()sin( Proof:
l.h.s. )sin()sin( yxyx )sincoscos(sin)sincoscos(sin yxyxyxyx
yxyxyxyx sincoscossinsincoscossin
yx sincos2... shr
( formulae (1) and (2) )
Trig Addition Formulae
Have a go
Relay race
In groups of 3-4
Exam questions
Exam questions
Plenary
Step by step on whiteboards1.Expand sin(x+30) and cos(x+60)2.Simplify by using exact values
3.Look at the form you’re aiming for and rearrange
4.Use rearranged form and rearrange again to make single trig ratio
5.Find principal value and any symmetry or periodicity values
Double Angle Double Angle FormulaeFormulaeObjectives:
To recognise and learn the double angle formulae for Sin 2A, Cos 2A and Tan 2A.To apply the double angle formulae to solving trig
equations and proving trig identities.
Double Angle Identities (1)
sin (A + A) = sin A cos A + cos A sin A
sin (2A)
sin (2A) = 2 sin A cos A
sin (A + B) = sin A cos B + cos A sin B
Setting A = B
Double Angles Identities (2)
cos (A + A) = cos A cos A - sin A sin A
cos (2A)
cos (2A) = cos2 A - sin2 A
Since, sin2x + cos2x = 1cos (2A) = cos2 A – (1 - cos2x) = 2 cos2 A - 1
cos (A + B) = cos A cos B - sin A sin B
Setting A = B
cos (2A) = 1 - sin2x – sin2 A = 1 - 2 sin2 A
SUMMARY
The double angle formulae are:
AAA cossin22sin )1(
AAA 22 sincos2cos )2(
1cos2 2 A )2( a
A2sin21 )2( b
A
AA
2tan1
tan22tan
)3(
N.B. The formulae link any angle with double the angle.For example, they can be used for x2 xan
d x
2
xand
y32
3 yand
We use them • to solve equations
• to prove other identities• to integrate some functions
4 and
2
Activity:Trig double angle
Card match
Using double angle formulae to prove identities
We can use the double angle formulae to prove other identities involving multiple angles. For example:
3cos3 4cos 3cos
Solve the following equations for the given intervals. Give answers correct to the nearest whole degree where appropriate. Where radians are required, exact answers should be given.
Exercise
1.
2.
3.
3600 x
,1cos2cos ,0sin2sin3 xx
180180 x,0tan32tan2 xx
Solution:
ANS: 360,280,180,80,0x
0sin2sin3 xx
1. ,0sin2sin3 xx 3600 x
0sin)cossin2(3 xxx
0sincossin6 xxx
0)1cos6(sin xx0sin x
61cos xor
Solution:
ANS:2
,3
,3
,2
1cos2cos
1cos1cos2 2
0coscos2 2
0)1cos2(cos
or0cos 21cos
2. ,1cos2cos
Solution:
)tan( t
0tan3tan1
tan42
xx
x
or0tan 372tan
,0tan32tan2 xx
0)1(34 2 ttt
037 3 tt
3. 180180 x,0tan32tan2 xx
37tan
ANS: 123,57,0,57,123,180 x
0)37( 2 tt
Prove the following identities:
1.
2.
3.
)sin(coscos212sin2cos xxxxx
Exercise
AAA cos3cos43cos 3
cot
2cos1
2sin
1. Prove )sin(coscos212sin2cos xxxxx
Proof:
1cossin21cos2 2 xxx
)sin(coscos2 xxx
= r.h.s.
l.h.s. 12sin2cos xx
Solutions:
(double angle formulae)
= r.h.s.
2. Prove
cot
2cos1
2sin
Proof:
)sin21(1
cossin22
l.h.s.
2cos1
2sin
2sin211
cossin2
2sin2
cossin2
Solutions:
cot
(double angle formulae)
= r.h.s.
Solutions:
3. Prove
)cossin2(sin)1cos2(cos 2 AAAAA AAAA cossin2coscos2 23
AAAA cos)cos1(2coscos2 23 AAAA 33 cos2cos2coscos2
AA cos3cos4 3
(addition formula) AAAA 2sinsin2coscos
Proof:
(double angle formulae)
)1sin(cos 22 AA
)2cos(... AAshl
AAA cos3cos43cos 3
Activity:
True or falseworksheet
SUMMARY
BB cos)cos( BB sin)sin(
BA
BABA
tantan1
tantan)tan(
BABABA sincoscossin)sin(
BABABA sinsincoscos)cos(
You need to remember the following results.
The addition formulae are in your formula booklets and are written as
Notice that the cos formulae have opposite signs on the 2 sides.
Using the Addition Formulae e.g. Prove the following:
xyyxyx cossin2)sin()sin( Proof:
l.h.s. )sin()sin( yxyx )sincoscos(sin)sincoscos(sin yxyxyxyx
yxyxyxyx sincoscossinsincoscossin
yx sincos2... shr
( formulae (1) and (2) )
SUMMARY
The double angle formulae are:
AAA cossin22sin )1(
AAA 22 sincos2cos )2(
1cos2 2 A )2( a
A2sin21 )2( b
A
AA
2tan1
tan22tan
)3(
N.B. The formula link any angle with double the angle.
For example, they can by used for x2 xan
d x
2
xand
4 2and
y32
3 yand
We use them • to solve equations
• to prove other identities• to integrate some functions
Proof: l.h.s. = )2sin( AA
= r.h.s.
)cossin2(cos)sin21(sin 2 AAAAA AAAA 23 cossin2sin2sin
)sin1(sin2sin2sin 23 AAAA AAAA 33 sin2sin2sin2sin
AA 3sin4sin3
e.g. Prove that AAA 3sin4sin33sin
(addition formula) AAAA 2sincos2cossin (double angle formulae)
)1sin(cos 22 AA
SUMMARY
A2cosThe rearrangements of the double angle formulae for are
)2cos1(cos 212 AA
)2cos1(sin 212 AA
They are important in integration so you should either memorise them or be able to obtain them very quickly.