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.