30: Trig addition formulae © Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core Modules

30: Trig addition 30: Trig addition formulaeformulae

““Teach A Level Maths”Teach A Level Maths”

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Trig Addition Formulae

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Module C3

Edexcel

Module C4

MEI/OCROCR

Does ? 60sin30sin)6030sin(

So, 60sin30sin)6030sin(

We cannot simplify the brackets as we do in algebra because they don’t mean multiply.

90sin)6030sin(l.h.s. =

160sin30sin

r.h.s. =

BBAA cossin,cos,sin and

The result, however, is true for any size of angles.

We’ll find the formula for where A and B are in degrees and where

)sin( BA

The proof is complicated but you are not expected to remember it !

However, can be written in terms of )sin( BA

Consider this rectangle

Tilt the rectangle through an angle A.

Let PR = 1

We can now find

using a right angled triangle

)sin( BA

h )sin( BA

)90( A

)sin( BA

h =But

)sin( BA M

NM + MRh =But

)sin( BA

NM + MR = TQ +T

h =But MR

= TQ + MR

But, TQ = Aa sin

)sin( BA

h =But NM + MR

= TQ + MR

But, TQ = Aa sin and MR =

)sin( BA

h =But NM + MR

= TQ + MR

But, TQ = Aa sin Abcosand MR =

)sin( BA

AbAah cossin

h =But NM + MR

AbAah cossin

)sin( BA

Also Bcos1a

)sin( BA

Also Bcos1a

Bsin b1b

AbAah cossin

)sin( BA

ABABh cossinsincos So,

Also Bcos1a

Bsin b

hBA )sin( BABABA sincoscossin)sin(

AbAah cossin

BABAh sincoscossin

xy sinxy cos

BABABA sincoscossin)sin(

Before we find the other addition formulae we need to notice 4 relationships between some of the trig ratios.

)cos( B

)sin( B

Bcos Bsin

BABABA sincoscossin)sin( )1(

xy cos

xy sin

)90sin( A

90 - A

)90sin(cos AA

xy cosxy sin

)90cos( A

90 - A

)90cos(sin AA

Now we can easily find 5 more addition formulae Replace B by (–B) in (1) : )sin(cos)cos(sin))(sin( BABABA

We now have

BB cos)cos( BB sin)sin( )90sin(cos AA AA sin)90cos(

)sin(coscossin)sin( BABABA BABABA sincoscossin)sin( )2(

BABABA sincoscossin)sin( and )1(

Trig Addition Formulae We now

haveBB cos)cos( BB sin)sin(

)90sin(cos AA AA sin)90cos(

BABABA sincoscossin)sin( )1(and

Replace A by ( 90 A ) in (2) :

BABABA sin)90cos(cos)90sin()90sin(

BABABA sinsincoscos))(90sin( BABABA sinsincoscos)cos(

)3(BABABA sinsincoscos)cos(

Exercise: Use (3) to find a formula for )cos( BA

Replace B by ( B ) in (3) :

)sin(sin)cos(cos)cos( BABABA

)sin(sincoscos)cos( BABABA BABABA sinsincoscos)cos( )4(

BABABA sinsincoscos)cos( )3(

BABABA sinsincoscos)cos(

These formulae are true for all values of A and B so they are identities. They should be written with identity signs.

sinsincoscos

sincoscossin

)sin()tan(

Divide numerator and denominator by :

BAcoscos

tantan1

tantan

)tan( BA

)tan( BA Formula for :

BAcoscos

BAcoscos BAcoscos

sinsincoscos

sincoscossin

BAcoscos BAcoscos

BAcoscos

tantan1

tantan

)5()tan( BA

Exercise: Using this formula, or otherwise, find a formula for )tan( BA Solution:

Replace B by ( B ) in (5) :

)tan(tan1

)tan(tan)tan(

tantan1

tantan)tan(

By dividing by we get

)sin()sin( BB )cos()cos( BB BB tan)tan(

OR: Use the method used to find formula (5)

SUMMARY

BB cos)cos( BB sin)sin(

tantan1

tantan)tan(

You need to remember the following results.

Check whether the addition formulae are in your formulae booklets. If so, they may be 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.

Using the Addition Formulae

Solution:

)4590sin(135sin

You will need your formulae booklets for the rest of this presentation and all the remaining

trig work.

BABABA sincoscossin)sin( Using

45sin90cos45cos90sin

We can rationalise the surd by multiplying numerator and denominator by2 2

e.g. 1 Find the exact value of simplifying the answer

135sin

)4590sin(

Using the Addition Formulae e.g. 2 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 Exercise

1. Simplifying the answers as much as possible, find exact values for:

75cos (b) 105sin (c) 15tan

2. Prove the following:yxyxyx sinsin2)cos()cos( (a)

(b) )sin()tan(tancoscos yxyxyx

(c) yxyx

yxtantan

coscos

You can assume some, or all, of the following: ,45sin45cos

2360sin30cos

2130sin60cos and

30sin45sin30cos45cos)3045cos(75cos

(b) 45sin60cos45cos60sin)4560sin(105sin

Solutions:

,45sin45cos2

1 2360sin30cos

2130sin60cos and

We can multiply numerator and denominator by to rationalise the surds.

)13)(13(

)13)(13(15tan

Solutions: ,45sin45cos

2360sin30cos

2130sin60cos and

145cos

45sin45tan

115tan

1315tan

Multiply numerator and denominator by

3Rationalise the surds 13

(c)30tan45tan1

30tan45tan)3045tan(

30sin30tan

yxyxyx sinsin2)cos()cos( 2(a) Prove

Solutions:

Proof: l.h.s. )cos()cos( yxyx

)sinsincos(cos)sinsincos(cos yxyxyxyx

yxyxyxyx sinsincoscossinsincoscos

yx sinsin2

... shr

Solutions:

Proof:

l.h.s. )tan(tancoscos yxyx

sincoscos

... shr

(b) )sin()tan(tancoscos yxyxyx

sincoscos

yxxy sincossincos )sin( yx using formula

sintan

coscos

sincos

coscos

cossin

Solutions:

Proof:

coscos

sincoscossin

... shr

(c) yxyx

yxtantan

coscos

using formula (1):

l.h.s. yx

coscos

yx tantan A

sintan

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.

Trig Addition Formulae SUMMAR

BB cos)cos( BB sin)sin(

tantan1

tantan)tan(

You need to remember the following results.

Check whether the addition formulae are in your formulae booklets. If so, they may be 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

30: Trig addition formulae © Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core Modules

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