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

Trig Addition Formulae "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages" Module C3 Edexcel Module C4 AQA MEI/OCROCR

Trig Addition Formulae Does ? and So, We cannot simplify the brackets as we do in algebra because they don’t mean multiply. l.h.s. = r.h.s. =

Trig Addition Formulae 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 The proof is complicated but you are not expected to remember it ! However, can be written in terms of

Trig Addition Formulae Consider this rectangle Tilt the rectangle through an angle A. Let PR = 1 We can now find using a right angled triangle 1 B a b R Q S P R Q P S 1 a b B A

Trig Addition Formulae Q P S 1 B A N a b R h A

Q P S 1 B A N a b R h h = But A

Trig Addition Formulae Q P S 1 B A N a b R h M NM +MRh = But A

Trig Addition Formulae Q P S 1 B A N a b R NM +MR= TQ + T h = But MR A M

Trig Addition Formulae Q P S 1 B A N a b R = TQ +MR But, TQ = h = But NM +MR M T A

Trig Addition Formulae Q P S 1 B A N a b R T M = TQ +MR But, TQ = and MR = h = But NM +MR A

Trig Addition Formulae Q P S 1 B A N a b R T M = TQ +MR But, TQ = and MR = h = But NM +MR A h

Trig Addition Formulae Q P S 1 B A N a b R M T Also

Trig Addition Formulae Q P S 1 B A N a b R M T Alsoand

Trig Addition Formulae Q P S 1 B A N a b R M So, Alsoand T

Trig Addition Formulae Before we find the other addition formulae we need to notice 4 relationships between some of the trig ratios. BB B BB B

Trig Addition Formulae A 90 - A Before we find the other addition formulae we need to notice 4 relationships between some of the trig ratios.

Trig Addition Formulae 90 - A A Before we find the other addition formulae we need to notice 4 relationships between some of the trig ratios.

Trig Addition Formulae Now we can easily find 5 more addition formulae Replace B by ( – B ) in (1) : We now have and

Trig Addition Formulae We now have and Replace A by ( 90  A ) in (2) :

Trig Addition Formulae We now have and Exercise: Use (3) to find a formula for

Trig Addition Formulae We now have and Replace B by (  B ) in (3) :

Trig Addition Formulae We now have and These formulae are true for all values of A and B so they are identities. They should be written with identity signs.

Trig Addition Formulae We now have and

Trig Addition Formulae Divide numerator and denominator by : Formula for :

Trig Addition Formulae Exercise: Using this formula, or otherwise, find a formula for Solution: Replace B by (  B ) in (5) : By dividing by we get so, OR: Use the method used to find formula (5)

Trig Addition Formulae SUMMARY 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.

Trig Addition Formulae Using the Addition Formulae Solution: You will need your formulae booklets for the rest of this presentation and all the remaining trig work. Using We can rationalise the surd by multiplying numerator and denominator by e.g. 1 Find the exact value of simplifying the answer

Trig Addition Formulae Using the Addition Formulae e.g. 2 Prove the following: Proof: l.h.s. ( formulae (1) and (2) )

Trig Addition Formulae Exercises (a) 1. Simplifying the answers as much as possible, find exact values for: (b)(c) 2. Prove the following: (a) (b) (c) You can assume some, or all, of the following: and

Trig Addition Formulae 1(a) (b) Solutions: and We can multiply numerator and denominator by to rationalise the surds.

Trig Addition Formulae Solutions: and Multiply numerator and denominator by Rationalise the surds (c) and

Trig Addition Formulae 2(a) Prove Solutions: Proof: l.h.s. ( formulae (3) and (4) )

Trig Addition Formulae Solutions: Proof: l.h.s. (b) using formula (2):

Trig Addition Formulae Solutions: Proof: (c) using formula (1 ): l.h.s.

Trig Addition Formulae

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 SUMMARY 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.

Trig Addition Formulae Using the Addition Formulae e.g. Prove the following: Proof: l.h.s. ( formulae (1) and (2) )