Introductionβ’ This chapter extends your knowledge of
Trigonometrical identities
β’ You will see how to solve equations involving combinations of sin, cos and tan
β’ You will learn to express combinations of these as a transformation of a single graph
Further Trigonometric Identities and their Applications
You need to know and be able to use the addition
formulae
7A
Q
P
N
11
AB
By GCSE Trigonometry:
O
So the coordinates of P are:
M So the coordinates of Q
are:
Q
P
ππππ΄βπ
πππ΅
Further Trigonometric Identities and their Applications
You need to know and be able to use the addition
formulae
7A
ππ2=ΒΏΒΏ
ππ2=ΒΏ(πΆπ π 2 π΄β2πΆππ π΄πΆππ π΅+πΆππ 2π΅)+(πππ2 π΄β2ππππ΄ππππ΅+πππ2π΅)
ππ2=ΒΏ(πΆπ π 2 π΄+πππ2 π΄) β2 (πΆππ π΄πΆππ π΅+ππππ΄ππππ΅)+(πΆππ 2π΅+πππ2π΅)
ππ2=ΒΏ2β2(πΆππ π΄πΆππ π΅+ππππ΄ππππ΅)
Multiply out the brackets
Rearrange
β‘ 1
Further Trigonometric Identities and their Applications
You need to know and be able to use the addition
formulae
7A
Q
P
N
11
A
B
O M
You can also work out PQ using the triangle
OPQ:
B - A1
1
2bcCosA 2Cos(B - A)
Q
P
2Cos(B - A) 2Cos(A - B)
Sub in the values
Group terms
Cos (B β A) = Cos (A β B) eg) Cos(60) = Cos(-60)
Further Trigonometric Identities and their Applications
You need to know and be able to use the addition
formulae
7A
2Cos(A - B)ππ2=ΒΏ2β2(πΆππ π΄πΆππ π΅+ππππ΄ππππ΅)
2β2(πΆππ π΄πΆππ π΅+ππππ΄ππππ΅) 2Cos(A - B)β2 (πΆππ π΄πΆππ π΅+ππππ΄ππππ΅) 2Cos(A - B)
πΆππ π΄πΆππ π΅+ππππ΄ππππ΅ Cos(A - B)Subtract 2 from both
sides
Divide by -2
Cos(A - B) = CosACosB + SinASinBCos(A + B) = CosACosB - SinASinB
Further Trigonometric Identities and their Applications
You need to know and be able to use the addition
formulae
7A
Cos(A - B) β‘ CosACosB + SinASinBCos(A + B) β‘ CosACosB - SinASinBSin(A + B) β‘ SinACosB + CosASinBSin(A - B) β‘ SinACosB - CosASinB
Further Trigonometric Identities and their Applications
You need to know and be able to use the addition
formulae
Show that:
7A
Cos(A - B) = CosACosB + SinASinBCos(A + B) = CosACosB - SinASinB
Sin(A + B) = SinACosB + CosASinBSin(A - B) = SinACosB - CosASinB
Tan (A + B) Tan ΞΈ
Tan (A+B) Tan (A+B)
Tan (A+B) Tan (A+B) TanA+ TanB1 - TanATanB
Rewrite
Divide top and bottom by CosACosB
Simplify each Fraction
Further Trigonometric Identities and their Applications
You need to know and be able to use the addition
formulae
7A
Cos(A - B) β‘ CosACosB + SinASinBCos(A + B) β‘ CosACosB - SinASinBSin(A + B) β‘ SinACosB + CosASinBSin(A - B) β‘ SinACosB - CosASinBTan (A + B)
Tan (A - B) You may be asked to
prove either of the Tan identities using the Sin
and Cos ones!
Further Trigonometric Identities and their Applications
You need to know and be able to use the addition
formulae
Show, using the formula for Sin(A β B), that:
7A
Cos(A - B) β‘ CosACosB + SinASinBCos(A + B) β‘ CosACosB - SinASinBSin(A + B) β‘ SinACosB + CosASinBSin(A - B) β‘ SinACosB - CosASinBTan (A + B)
Tan (A - B)
πππ15=β6ββ24
πππ15=πππ(45β30)
Sin(A - B) β‘ SinACosB - CosASinBSin(45 - 30) β‘ Sin45Cos30 β Cos45Sin30
Sin(45 - 30) β‘β22
β32
β2212Γ Γβ
Sin(45 - 30) β‘β64
β24β
Sin(15) β‘ β6ββ24
A=45, B=30
These can be written as surds
Multiply each pair
Group the fractions up
Further Trigonometric Identities and their Applications
You need to know and be able to use the addition
formulae
Given that:
Find the value of:
7A
< A < 270ΛCosTan(A+B)
Tan (A + B)
A
B
35
4
12
13
ππππ΄=ππππ»π¦π
ππππ΄=β 35
πΆππ π΅=π΄πππ»π¦π 5
ππππ΄=ππππ΄ππ
ππππ΄=34
ππππ΅=ππππ΄ππ
ππππ΅=512πΆππ π΅=β 1213
ππππ΅=β 512
Use Pythagorasβ to find the missing side (ignore negatives)Tan is positive in the range 180Λ -
270Λ
Use Pythagorasβ to find the missing side (ignore negatives)Tan is negative in the range 90Λ -
180Λ
90 180
270 360y = TanΞΈ
Further Trigonometric Identities and their Applications
You need to know and be able to use the addition
formulae
Given that:
Find the value of:
7A
< A < 270ΛCosTan(A+B)
Tan (A + B)
Tan (A + B)
ππππ΄=34 ππππ΅=β 512
Tan (A + B) Tan (A + B) Tan (A + B)
Tan (A + B)
Substitute in TanA and TanB
Work out the Numerator and Denominator
Leave, Change and Flip
Simplify
Although you could just type the whole thing into your calculator, you still need to show the stages for
the workings marksβ¦
Further Trigonometric Identities and their Applications
You need to know and be able to use the addition formulaeGiven that:
Express Tanx in terms of Tanyβ¦
7A
2π ππ (π₯+ π¦ )=3πππ (π₯β π¦ )
2π ππ (π₯+ π¦ )=3πππ (π₯β π¦ )
2(π πππ₯πππ π¦+πππ π₯π πππ¦ )ΒΏ3 (πππ π₯πππ π¦+π πππ₯π πππ¦ )
2π πππ₯πππ π¦+2πππ π₯π πππ¦ΒΏ3πππ π₯πππ π¦+3 π πππ₯π πππ¦2π πππ₯πππ π¦+2πππ π₯π πππ¦ΒΏ3πππ π₯πππ π¦+3 π πππ₯π πππ¦πππ π₯πππ π¦ πππ π₯πππ π¦ πππ π₯πππ π¦ πππ π₯πππ π¦
2 π‘πππ₯+2 π‘πππ¦ΒΏ3+3 π‘πππ₯π‘πππ¦2 π‘πππ₯β3 π‘πππ₯π‘πππ¦ΒΏ3β2π‘πππ¦π‘πππ₯ (2β3 π‘πππ¦ )ΒΏ3β2π‘πππ¦
π‘πππ₯ΒΏ3β2π‘πππ¦2β3 π‘πππ¦
Rewrite the sin and cos parts
Multiply out the brackets
Divide all by cosxcosy
Simplify
Subtract 3tanxtanySubtract 2tany
Factorise the left side
Divide by (2 β 3tany)
Further Trigonometric Identities and their Applications
You can express sin2A, cos 2A and tan2A in terms of angle A,
using the double angle formulae
7B
Sin(A + B) β‘ SinACosB + CosASinBSin(A + A) β‘ SinACosA + CosASinASin2A β‘ 2SinACosA
Replace B with A
Simplify
Sin2A β‘ 2SinACosASin4A β‘ 2Sin2ACos2ASin2A β‘ SinACosA
Sin60 β‘ 2Sin30Cos303Sin2A β‘ 6SinACosA
Γ· 2
2A 4A
x 3 2A = 60
Further Trigonometric Identities and their Applications
You can express sin2A, cos 2A and tan2A in terms of angle A,
using the double angle formulae
7B
Cos(A + B) β‘ CosACosB - SinASinBCos(A + A) β‘ CosACosA - SinASinA
Cos2A β‘ CoReplace B with A
Simplify
Cos2A β‘ CoCos2A β‘ (1 Cos2A β‘ Co1 - Co
Cos2A β‘ 1 Cos2A β‘ 2CoReplace Sin2A with (1 β Cos2A)Replace Cos2A with (1 β Sin2A)
Further Trigonometric Identities and their Applications
You can express sin2A, cos 2A and tan2A in terms of angle A,
using the double angle formulae
7B
Replace B with A
Simplify
Tan (A + B) Tan (A + A)
Tan 2A
Tan 2A Tan 60 Tan 2A
2Tan 2A Tan A Γ· 2 2A = 60
x 22A = A
Further Trigonometric Identities and their Applications
You can express sin2A, cos 2A and tan2A in terms of angle A,
using the double angle formulae
Rewrite the following as a single Trigonometric function:
7B
2π ππ π2 πππ
π2 πππ π
πππ 2πβ‘2 π ππππππ πππππβ‘2 π ππ π2 πππ
π2
2π ππ π2 πππ
π2 πππ π
ΒΏ π ππππππ π
ΒΏ12 π ππ2π
2ΞΈ ΞΈ
Replace the first part
Rewrite
Further Trigonometric Identities and their Applications
You can express sin2A, cos 2A and tan2A in terms of angle A,
using the double angle formulae
Show that:
Can be written as:
7B
1+πππ 4π
πΆππ 2 πβ‘2πππ 2πβ1 Double the angle parts
2πππ 22π
πΆππ 4 πβ‘2πππ 22πβ1
1+πππ 4πΒΏ1+(2πππ 22πβ1)
ΒΏ2πππ 22 π
Replace cos4ΞΈ
The 1s cancel out
Further Trigonometric Identities and their Applications
You can express sin2A, cos 2A and tan2A in terms of angle A,
using the double angle formulae
Given that:
Find the exact value of:
7B
πππ π₯=34 180 Λ<π₯<360 Λ
π ππ2π₯
x
πΆππ π₯=π΄πππ»π¦π
πΆππ π₯=34
ππππ₯=ππππ»π¦π
ππππ₯=β74
Use Pythagorasβ to find the missing side (ignore negatives)
Cosx is positive so in the range 270 - 360
3
β74
Therefore, Sinx is negative
90 180
270 360
y = SinΞΈ
y = CosΞΈ
ππππ₯=β β74
Sin2x β‘ 2SinxCosxSin2x = 2Sin2x =
Sub in Sinx and Cosx
Work out and leave in surd form
Further Trigonometric Identities and their Applications
You can express sin2A, cos 2A and tan2A in terms of angle A,
using the double angle formulae
Given that:
Find the exact value of:
7B
πππ π₯=34 180 Λ<π₯<360 Λ
π‘ππ2π₯
x
πΆππ π₯=π΄πππ»π¦π
πΆππ π₯=34
ππππ₯=ππππ΄ππ
ππππ₯=β73
Use Pythagorasβ to find the missing side (ignore negatives)
Cosx is positive so in the range 270 - 360
3
β74
Therefore, Tanx is negative
90 180
270 360y = CosΞΈ
ππππ₯=β β73
Sub in Tanx
Work out and leave in surd form90 18
0270 360
y = TanΞΈ
Tan 2x Tan 2x
πππ2 π₯=β3β7
Further Trigonometric Identities and their Applications
The double angle formulae allow you to solve more
equations and prove more identities
Prove the identity:
7C
π‘ππ2πβ‘ 2πππ‘ πβ π‘πππ
π‘ππ2πβ‘ 2π‘ππ π1βπ‘ππ2π
π‘ππ2πβ‘
2 π‘ππππ‘πππ
1π‘πππ β
π‘ππ2ππ‘πππ
π‘ππ2πβ‘ 2πππ‘ πβπ‘πππ
Divide each part by tanΞΈ
Rewrite each part
Further Trigonometric Identities and their Applications
The double angle formulae allow you to solve more
equations and prove more identities
By expanding:
Show that:
7C
π ππ (2π΄+π΄)
Replace A and B
π ππ (3 π΄)β‘3 π πππ΄β4 π ππ3 π΄
π ππ (π΄+π΅ )β‘π πππ΄πππ π΅+πππ π΄π πππ΅
π ππ (2π΄+π΄ )β‘π ππ 2π΄πππ π΄+πππ 2 π΄π πππ΄
π ππ (3 π΄)β‘(2 π πππ΄πππ π΄)πππ π΄+(1β2π ππ2 π΄) π πππ΄
π ππ (3 π΄)β‘2π πππ΄πππ 2 π΄+π πππ΄β2π ππ3π΄
π ππ (3 π΄)β‘2π πππ΄ (1βπ ππ2 π΄)+π πππ΄β2π ππ3π΄
π ππ (3 π΄)β‘2π πππ΄β2π ππ3π΄+π πππ΄β2π ππ3π΄
π ππ (3 π΄)β‘3 π πππ΄β4π ππ3 π΄
Replace Sin2A and
Cos 2AMultiply
outReplace cos2A
Multiply out
Group like terms
Further Trigonometric Identities and their Applications
The double angle formulae allow you to solve more
equations and prove more identities
Given that:
Eliminate ΞΈ and express y in terms of xβ¦
7C
π₯=3 π πππ π¦=3β4 πππ 2 πand
π₯=3 π ππππ₯3=π πππ
π¦=3β4 πππ 2 π3βπ¦4 =πππ 2π
Divide by 3
Subtract 3, divide by 4Multiply by -1
πππ 2π=1β2π ππ2π
3βπ¦4 =ΒΏ1β2( π₯3 )
2
3β π¦=ΒΏ4β8( π₯3 )2
β π¦=ΒΏ1β8 (π₯3 )2
π¦=ΒΏ8 (π₯3 )2
β1
Replace Cos2ΞΈ and SinΞΈ
Multiply by 4
Subtract 3
Multiply by -1
Further Trigonometric Identities and their Applications
The double angle formulae allow you to solve more
equations and prove more identities
Solve the following equation in the range stated:
(All trigonometrical parts must be in terms x, rather than 2x)
7C
3πππ 2π₯βπππ π₯+2=00 Β° β€ π₯β€360 Β°
3πππ 2π₯βπππ π₯+2=0
3 (2πππ 2π₯β1)βπππ π₯+2=0
6πππ 2π₯β3βπππ π₯+2=0
6πππ 2π₯βπππ π₯β1=0
Replace cos2x
Multiply out the bracket
Group terms
(3πππ π₯+1)(2πππ π₯β1)=0
πππ π₯=β 13 πππ π₯=12or
Factorise
90 180
270 360
y = CosΞΈ1
2β 13
π₯=πππ β1(β 13 ) π₯=πππ β1( 12 )π₯=109.5 Β°,250.5 Β°π₯=60 Β°,300 Β°
π₯=60 Β° ,109.5 Β° ,250.5 Β° ,300 Β°
Solve both pairs
Remember to find additional answers!
Further Trigonometric Identities and their Applications
You can write expressions of the form acosΞΈ + bsinΞΈ,
where a and b are constants, as a sine or cosine function
only
Show that:
Can be expressed in the form:
So:
7D
3 π πππ₯+4πππ π₯
π π ππ(π₯+Ξ± )
π π ππ(π₯+Ξ± )ΒΏπ π πππ₯πππ Ξ±+π πππ π₯π ππ Ξ±3 π πππ₯+4πππ π₯ΒΏπ π πππ₯πππ Ξ±+π πππ π₯π ππ Ξ±π πππ Ξ±=3 π π ππΞ±=4
πππ Ξ±= 3π π ππΞ±= 4
π Ξ±
π
3
4(ππ» )( π΄π» )So in the triangle, the Hypotenuse is
Rβ¦π =β32+42 π =5
πππ Ξ±= 3π
πππ Ξ±=35
Ξ±=πππ β 1 35
Ξ±=53.1Β°
Replace with the expression
Compare each term β they must be equal!
R = 5
Inverse Cos
Find the smallest value in the acceptable range given
3 π πππ₯+4πππ π₯ΒΏ5sin (π₯+53.1Β° )
Further Trigonometric Identities and their Applications
You can write expressions of the form acosΞΈ + bsinΞΈ,
where a and b are constants, as a sine or cosine function
only
Show that you can express:
In the form:
So:
7D
π πππ₯ββ3πππ π₯
π π ππ(π₯βΞ±)
π π ππ(π₯βΞ±)ΒΏπ π πππ₯πππ πΌβπ πππ π₯π πππΌπ πππ₯ββ3πππ π₯ΒΏπ π πππ₯πππ πΌβπ πππ π₯π πππΌπ πππ πΌ=1 π π πππΌ=β3
π =β12+(β3 )2 π =2
π πππ πΌ=12πππ πΌ=1πππ πΌ=
12
πΌ=πππ β1 12
πΌ=π3
R = 2
Divide by 2
Inverse cos
Find the smallest value in the
acceptable range
Replace with the expression
Compare each term β they must be equal!
π πππ₯ββ3πππ π₯ΒΏ2sin (π₯β π3 )
Further Trigonometric Identities and their Applications
You can write expressions of the form acosΞΈ + bsinΞΈ,
where a and b are constants, as a sine or cosine function
only
Show that you can express:
In the form:
So:
7D
π πππ₯ββ3πππ π₯
π π ππ(π₯βΞ±)
π πππ₯ββ3πππ π₯ΒΏ2sin (π₯β π3 )
Sketch the graph of:π πππ₯ββ3πππ π₯= Sketch the graph of:2sin (π₯β π3 )
Ο/2 Ο 3Ο/2 2Ο
1
-1
Ο/2 Ο 3Ο/2 2Ο
1
y=sin π₯
y=sin (π₯β π3 )
Ο/2 Ο 3Ο/2 2Ο
1
-1
y=2sin (π₯β π3 )
Ο/34Ο/3-1
2
-2
Start out with sinx
Translate Ο/3 units
right
Vertical stretch, scale
factor 2Ο/34Ο/3
2sin (β π3 )ΒΏββ3 At the y-
intercept, x = 0
Further Trigonometric Identities and their Applications
You can write expressions of the form acosΞΈ + bsinΞΈ,
where a and b are constants, as a sine or cosine function
only
Express:
in the form:
So:
7D
2πππ π+5π πππ
π πππ (πβπΌ)
π πππ (πβπΌ)ΒΏπ πππ ππππ πΌ+π π ππππ πππΌ2πππ π+5π πππΒΏπ πππ ππππ πΌ+π π ππππ πππΌπ πππ πΌ=2 π π πππΌ=5
Replace with the expression
Compare each term β they must be equal!
π =β22+52 π =β29
π πππ πΌ=2β29πππ πΌ=2πππ πΌ=
2β29
πΌ=πππ β1 2β29
πΌ=68.2
R = β29
Divide by β29
Inverse cos
Find the smallest value in the
acceptable range
2πππ π+5π πππΒΏβ29cos (πβ68.2)
Further Trigonometric Identities and their Applications
You can write expressions of the form acosΞΈ + bsinΞΈ,
where a and b are constants, as a sine or cosine function
only
Solve in the given range, the following equation:
7D
2πππ π+5π πππ=30 Β°<π<360 Β°
2πππ π+5π πππ=β29cos (πβ68.2)
β29cos (πβ68.2)=30 Β°<π<360 Β°
We just showed that the original equation can be rewrittenβ¦
Hence, we can solve this equation instead!
β68.2 Β°<πβ68.2<291.2Β°Remember to
adjust the range for (ΞΈ β
68.2)
β29cos (πβ68.2)=3
cos (πβ68.2)= 3β29
πβ68.2=πππ β 1 3β29
πβ68.2=56.1,β56.1,303.9
π=12.1,124.3
Divide by β29
Inverse Cos
Remember to work out other values in the
adjusted range
Add 68.2 (and put in order!)
90 180
270 360
y = CosΞΈ
-90
56.1
-56.1
303.9
Further Trigonometric Identities and their Applications
You can write expressions of the form acosΞΈ + bsinΞΈ,
where a and b are constants, as a sine or cosine function
only
Find the maximum value of the following expression, and the smallest positive value of ΞΈ at
which it arises:
7D
12πππ π+5 π πππ
π πππ (πβπΌ)ΒΏπ πππ ππππ πΌ+π π ππππ πππΌ12πππ π+5 π πππΒΏπ πππ ππππ πΌ+π π ππππ πππΌ
π πππ πΌ=12 π π πππΌ=5
Replace with the expression
Compare each term β they must be equal!
π =β122+52π =13
π πππ πΌ=1213πππ πΌ=12πππ πΌ=
1213
πΌ=πππ β1 1213
πΌ=22.6
R = 13
Divide by 13
Inverse cos
Find the smallest value in the
acceptable range
ΒΏ13cos (πβ22.6 )
13cos (πβ22.6)13(1)πππ₯=13πβ22.6=0π=22.6
Max value of cos(ΞΈ - 22.6) =
1Overall maximum
therefore = 13Cos peaks at 0
ΞΈ = 22.6 gives us 0
Rcos(ΞΈ β Ξ±) chosen as it gives us the same
form as the expression
Further Trigonometric Identities and their Applications
You can write expressions of the form acosΞΈ + bsinΞΈ,
where a and b are constants, as a sine or cosine function
only
7D
ππ ππ πΒ±ππππ π
ππππ πΒ±ππ πππ
π π ππ (πΒ±πΌ )
π πππ (πβπΌ )
Whichever ratio is at the start, change the expression into a function of that (This makes solving problems
easier)Remember to get the + or β signs the correct way
round!
Further Trigonometric Identities and their Applications
You can express sums and differences of sines and cosines as products of sines and cosines by
using the βfactor formulaeβ
7E
π πππ+π πππ=2 π ππ( π+π2 )πππ ( πβπ2 )
π πππβπ πππ=2πππ ( π+π2 )π ππ( πβπ2 )
πππ π+πππ π=2πππ ( π+π2 )πππ (π βπ2 )
πππ πβπππ π=β2π ππ( π+π2 )π ππ( π βπ2 )
You get given all these in the formula booklet!
Further Trigonometric Identities and their Applications
You can express sums and differences of sines and cosines as products of sines and cosines by
using the βfactor formulaeβ
7E
π πππ+π πππ=2 π ππ( π+π2 )πππ ( πβπ2 )
π πππβπ πππ=2πππ ( π+π2 )π ππ( πβπ2 )
πππ π+πππ π=2πππ ( π+π2 )πππ (π βπ2 )
πππ πβπππ π=β2π ππ( π+π2 )π ππ( π βπ2 )
Using the formulae for Sin(A + B) and Sin (A β B), derive the result that:
π πππ+π πππ=2 π ππ( π+π2 )πππ ( πβπ2 )
πππ ( π΄+π΅ )=π πππ΄πππ π΅+πππ π΄π πππ΅πππ ( π΄βπ΅ )=π πππ΄πππ π΅βπππ π΄π πππ΅πππ ( π΄+π΅ )+π ππ (π΄βπ΅)=2 π πππ΄πππ π΅
π΄+π΅=ππ΄βπ΅=π2 π΄=π+ππ΄=
π+π2
π΄+π΅=ππ΄βπ΅=π2π΅=πβππ΅=
π βπ2
Add both sides together (1 + 2)
1)
2)
Let (A+B) = P Let (A-B) = Q
1)
2)
1)
2)1 + 2
Divide by
2
1 - 2
Divide by
2
πππ ( π΄+π΅ )+π ππ (π΄βπ΅)=2 π πππ΄πππ π΅
ππππ+ππππΒΏ2 π ππ( π+π2 ) ( πβπ2 )πππ
Further Trigonometric Identities and their Applications
You can express sums and differences of sines and cosines as products of sines and cosines by
using the βfactor formulaeβ
7E
π πππ+π πππ=2 π ππ( π+π2 )πππ ( πβπ2 )
π πππβπ πππ=2πππ ( π+π2 )π ππ( πβπ2 )
πππ π+πππ π=2πππ ( π+π2 )πππ (π βπ2 )
πππ πβπππ π=β2π ππ( π+π2 )π ππ( π βπ2 )
Show that:
π ππ105βπ ππ15= 1β2
π πππβπ πππ=2πππ ( π+π2 )π ππ( πβπ2 )
π ππ105βπ ππ15=2πππ ( 105+152 )π ππ(105β152 )π ππ105βπ ππ15=2πππ 60 π ππ 45
π ππ105βπ ππ15=ΒΏ2Γ12
1β2Γ
π ππ105βπ ππ15=ΒΏ1β2
P = 105 Q
= 15Work out the fraction parts
Sub in values for Cos60 and
Sin45Work out the
right hand side
Further Trigonometric Identities and their Applications
You can express sums and differences of sines and cosines as products of sines and cosines by
using the βfactor formulaeβ
7E
π πππ+π πππ=2 π ππ( π+π2 )πππ ( πβπ2 )
π πππβπ πππ=2πππ ( π+π2 )π ππ( πβπ2 )
πππ π+πππ π=2πππ ( π+π2 )πππ (π βπ2 )
πππ πβπππ π=β2π ππ( π+π2 )π ππ( πβπ2 )
Solve in the range indicated:π ππ4 πβπ ππ3 π=0 0β€ πβ€π
π πππβπ πππ=2πππ ( π+π2 )π ππ( πβπ2 )
π ππ4 πβπ ππ3 π=2πππ ( 4 π+3π2 )π ππ( 4 πβ3π2 )
π ππ4 πβπ ππ3 π=2πππ ( 7π2 )π ππ(π2 )2πππ ( 7 π2 )π ππ( π2 )=0πππ ( 7 π2 )=07π2 =πππ β 100β€ 7π2 β€
7π2
0β€ πβ€π
7π2 =
π2 ,3π2 ,5π2 ,7π2
π=π7 ,3π7 ,5π7 ,π
P = 4ΞΈ Q = 3ΞΈWork out
the fractions
Set equal to 0
Either the cos or sin part must equal 0β¦
Inverse cos
Solve, remembering to take into account the different range
Once you have all the values from 0-2Ο, add 2Ο to them to obtain equivalentsβ¦
Multiply by 2 and divide by 7
Adjust the range
Ο/2 Ο 3Ο/2 2Οy = CosΞΈ
0
Further Trigonometric Identities and their Applications
You can express sums and differences of sines and cosines as products of sines and cosines by
using the βfactor formulaeβ
7E
π πππ+π πππ=2 π ππ( π+π2 )πππ ( πβπ2 )
π πππβπ πππ=2πππ ( π+π2 )π ππ( πβπ2 )
πππ π+πππ π=2πππ ( π+π2 )πππ (π βπ2 )
πππ πβπππ π=β2π ππ( π+π2 )π ππ( πβπ2 )
Solve in the range indicated:π ππ4 πβπ ππ3 π=0 0β€ πβ€π
π πππβπ πππ=2πππ ( π+π2 )π ππ( πβπ2 )
π ππ4 πβπ ππ3 π=2πππ ( 4 π+3π2 )π ππ( 4 πβ3π2 )
π ππ4 πβπ ππ3 π=2πππ ( 7π2 )π ππ(π2 )2πππ ( 7 π2 )π ππ( π2 )=0π ππ( π2 )=0π2 =π ππβ 100β€ π2 β€
π2
0β€ πβ€π
π2 =0
π=0
P = 4ΞΈ Q = 3ΞΈWork out
the fractions
Set equal to 0
Either the cos or sin part must equal 0β¦
Inverse sin
Solve, remembering to take into account the different range
Once you have all the values from 0-2Ο, add 2Ο to them to obtain equivalents
Multiply by 2
Adjust the range
Ο/2 Ο 3Ο/2 2Οy = SinΞΈ
0
Further Trigonometric Identities and their Applications
You can express sums and differences of sines and cosines as products of sines and cosines by
using the βfactor formulaeβ
Prove that:
7E
π πππ+π πππ=2 π ππ( π+π2 )πππ ( πβπ2 )
π πππβπ πππ=2πππ ( π+π2 )π ππ( πβπ2 )
πππ π+πππ π=2πππ ( π+π2 )πππ (π βπ2 )
πππ πβπππ π=β2π ππ( π+π2 )π ππ( πβπ2 )
π ππ (π₯+2 π¦ )+π ππ (π₯+π¦ )+π πππ₯πππ (π₯+2 π¦ )+πππ (π₯+π¦ )+πππ π₯
=π‘ππ (π₯+ π¦ )
π ππ (π₯+2 π¦ )+π ππ (π₯+π¦ )+π πππ₯In the numerator:
π ππ (π₯+2 π¦ )+π πππ₯
π πππ+π πππ=2 π ππ( π+π2 )πππ ( πβπ2 )
π ππ (π₯+2 π¦ )+π πππ₯=2π ππ( π₯+2 π¦+π₯2 )πππ ( π₯+2 π¦βπ₯
2 )ΒΏ2 π ππ (π₯+ π¦ )πππ π¦
ΒΏ2 π ππ (π₯+π¦ )πππ π¦+π ππ (π₯+π¦ )
ΒΏ π ππ (π₯+ π¦ )(2πππ π¦+1)
Ignore sin(x + y) for nowβ¦
Use the identity for adding 2 sines
P = x + 2y Q = x
Simplify Fractions
Bring back the sin(x + y) we ignored earlier
Factorise
Numerator: π ππ (π₯+ π¦ )(2πππ π¦+1)
Further Trigonometric Identities and their Applications
You can express sums and differences of sines and cosines as products of sines and cosines by
using the βfactor formulaeβ
Prove that:
7E
π πππ+π πππ=2 π ππ( π+π2 )πππ ( πβπ2 )
π πππβπ πππ=2πππ ( π+π2 )π ππ( πβπ2 )
πππ π+πππ π=2πππ ( π+π2 )πππ (π βπ2 )
πππ πβπππ π=β2π ππ( π+π2 )π ππ( πβπ2 )
π ππ (π₯+2 π¦ )+π ππ (π₯+π¦ )+π πππ₯πππ (π₯+2 π¦ )+πππ (π₯+π¦ )+πππ π₯
=π‘ππ (π₯+ π¦ )
πππ (π₯+2 π¦ )+πππ (π₯+π¦ )+πππ π₯In the
denominator:
πππ (π₯+2 π¦ )+πππ π₯
πππ π+πππ π=2πππ ( π+π2 )πππ (π βπ2 )
πππ (π₯+2 π¦ )+πππ π₯=2πππ ( π₯+2 π¦+π₯2 )πππ ( π₯+2 π¦βπ₯
2 )ΒΏ2πππ (π₯+ π¦ )πππ π¦
ΒΏ2πππ (π₯+π¦ )πππ π¦+πππ (π₯+π¦ )
ΒΏπππ (π₯+ π¦ )(2πππ π¦+1)
Ignore cos(x + y) for nowβ¦
Use the identity for adding 2 cosines
P = x + 2y Q = x
Simplify Fractions
Bring back the cos(x + y) we ignored earlier
Factorise
Numerator: π ππ (π₯+ π¦ )(2πππ π¦+1)
Denominator:
πππ (π₯+ π¦ )(2πππ π¦+1)
Further Trigonometric Identities and their Applications
You can express sums and differences of sines and cosines as products of sines and cosines by
using the βfactor formulaeβ
Prove that:
7E
π πππ+π πππ=2 π ππ( π+π2 )πππ ( πβπ2 )
π πππβπ πππ=2πππ ( π+π2 )π ππ( πβπ2 )
πππ π+πππ π=2πππ ( π+π2 )πππ (π βπ2 )
πππ πβπππ π=β2π ππ( π+π2 )π ππ( πβπ2 )
π ππ (π₯+2 π¦ )+π ππ (π₯+π¦ )+π πππ₯πππ (π₯+2 π¦ )+πππ (π₯+π¦ )+πππ π₯
=π‘ππ (π₯+ π¦ )
Numerator: π ππ (π₯+ π¦ )(2πππ π¦+1)
Denominator:
πππ (π₯+ π¦ )(2πππ π¦+1)
π ππ (π₯+2 π¦ )+π ππ (π₯+π¦ )+π πππ₯πππ (π₯+2 π¦ )+πππ (π₯+π¦ )+πππ π₯
ΒΏπ ππ(π₯+π¦)(2πππ π¦+1)πππ (π₯+π¦ )(2πππ π¦+1)
ΒΏπ ππ(π₯+ π¦)πππ (π₯+π¦ )
ΒΏ π‘ππ (π₯+ π¦ )
Replace the numerator and denominator
Cancel out the (2cosy + 1)
brackets
Use one of the identities from C2
Summaryβ’ We have extended the range of techniques
we have for solving trigonometrical equations
β’ We have seen how to combine functions involving sine and cosine into a single transformation of sine or cosine
β’ We have learnt several new identities
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