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Introduction
β’ 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
Q
By GCSE Trigonometry:
1
So the coordinates of P are:
B
P
1
A
O
M
N
So the coordinates of Q are:
ππππ΄ β ππππ΅
Q
P
7A
Further Trigonometric Identities and their
Applications
You need to know and be able to
use the addition formulae
ππ2 = (πΆππ π΄ β πΆππ π΅)2 + (ππππ΄ β ππππ΅)2
Multiply out
the brackets
ππ2 = (πΆππ 2 π΄ β 2πΆππ π΄πΆππ π΅ + πΆππ 2 π΅) + (πππ2 π΄ β 2ππππ΄ππππ΅ + πππ2 π΅)
Rearrange
ππ2 = (πΆππ 2 π΄ + πππ2 π΄) + (πΆππ 2 π΅ + πππ2 π΅) β 2(πΆππ π΄πΆππ π΅ + ππππ΄ππππ΅)
πΆππ 2 ΞΈ + πππ2 ΞΈ β‘ 1
ππ2 = 2 β 2(πΆππ π΄πΆππ π΅ + ππππ΄ππππ΅)
7A
Further Trigonometric Identities and their
Applications
You need to know and be able to
use the addition formulae
Q
You can also work out PQ using the triangle OPQ:
Q
P
1
1
B-A
B
P
1
A
O
1
M
N
π2 = π 2 + π 2 β 2bcCosA
Sub in the values
ππ2 = 12 + 12 β 2Cos(B - A)
Group terms
ππ2 = 2 β 2Cos(B - A)
ππ2 = 2 β 2Cos(A - B)
Cos (B β A) = Cos (A β B)
eg) Cos(60) = Cos(-60)
7A
Further Trigonometric Identities and their
Applications
You need to know and be able to
use the addition formulae
ππ2 = 2 β 2(πΆππ π΄πΆππ π΅ + ππππ΄ππππ΅)
ππ2 = 2 β 2Cos(A - B)
2 β 2(πΆππ π΄πΆππ π΅ + ππππ΄ππππ΅) = 2 β 2Cos(A - B)
β 2(πΆππ π΄πΆππ π΅ + ππππ΄ππππ΅) = β 2Cos(A - B)
πΆππ π΄πΆππ π΅ + ππππ΄ππππ΅ = Cos(A - B)
Subtract 2 from both
sides
Divide by -2
Cos(A - B) = CosACosB + SinASinB
Cos(A + B) = CosACosB - SinASinB
7A
Further Trigonometric Identities and their
Applications
You need to know and be able to
use the addition formulae
Cos(A - B) β‘ CosACosB + SinASinB
Cos(A + B) β‘ CosACosB - SinASinB
Sin(A + B) β‘ SinACosB + CosASinB
Sin(A - B) β‘ SinACosB - CosASinB
7A
Further Trigonometric Identities and their
Applications
You need to know and be able to
use the addition formulae
Sin(A + B) = SinACosB + CosASinB
Sin(A - B) = SinACosB - CosASinB
Tan (A+B) β‘
Tan (A+B) β‘
Cos(A - B) = CosACosB + SinASinB
Cos(A + B) = CosACosB - SinASinB
Tan (A+B) β‘
Show that:
Tan (A + B) β‘
Tan ΞΈ β‘
ππππ΄+ππππ΅
1βππππ΄ππππ΅
Tan (A+B) β‘
πππ(π΄+π΅)
πΆππ (π΄+π΅)
ππππ΄πΆππ π΅+πΆππ π΄ππππ΅
πΆππ π΄πΆππ π΅βππππ΄ππππ΅
ππππ΄πΆππ π΅ πΆππ π΄ππππ΅
+
πΆππ π΄πΆππ π΅ πΆππ π΄πΆππ π΅
πΆππ π΄πΆππ π΅ ππππ΄ππππ΅
β
πΆππ π΄πΆππ π΅ πΆππ π΄πΆππ π΅
TanA + TanB
1 - TanATanB
Rewrite
Divide top and
bottom by
CosACosB
Simplify each
Fraction
πππΞΈ
πΆππ ΞΈ
7A
Further Trigonometric Identities and their
Applications
You need to know and be able to
use the addition formulae
Cos(A - B) β‘ CosACosB + SinASinB
Cos(A + B) β‘ CosACosB - SinASinB
Sin(A + B) β‘ SinACosB + CosASinB
Sin(A - B) β‘ SinACosB - CosASinB
Tan (A + B) β‘
ππππ΄+ππππ΅
1βππππ΄ππππ΅
Tan (A - B) β‘
ππππ΄βππππ΅
1+ππππ΄ππππ΅
You may be asked to prove
either of the Tan
identities using the Sin and
Cos ones!
7A
Further Trigonometric Identities and their
Applications
You need to know and be able to
use the addition formulae
Cos(A + B) β‘ CosACosB - SinASinB
πππ15 = πππ(45 β 30)
Sin(A - B) β‘ SinACosB - CosASinB
A=45,
B=30
Cos(A - B) β‘ CosACosB + SinASinB
Sin(A + B) β‘ SinACosB + CosASinB
Sin(45 - 30) β‘ Sin45Cos30 β Cos45Sin30
Sin(A - B) β‘ SinACosB - CosASinB
ππππ΄+ππππ΅
Tan (A + B) β‘ 1βππππ΄ππππ΅
ππππ΄βππππ΅
Tan (A - B) β‘ 1+ππππ΄ππππ΅
Show, using the formula for Sin(A β B),
that:
πππ15 =
6β 2
4
Sin(45 - 30) β‘
Sin(45 - 30) β‘
Sin(15) β‘
2
3
×
β
2
2
6
β
4
6β 2
4
2
4
1
2
×
2
2
These can
be written
as surds
Multiply
each pair
Group the
fractions up
7A
Further Trigonometric Identities and their
Applications
You need to know and be able to
use the addition formulae
Given that:
SinA = β
3
5
CosB = β
πππ
π»π¦π
3
ππππ΄ = β
5
180Λ < A < 270Λ
12
13
ππππ΄ =
5
3
A
πππ
π΄ππ
ππππ΄ =
3
4
4
B = Obtuse
Use Pythagorasβ to find the missing side (ignore negatives)
Find the value of:
Tan is positive in the range 180Λ - 270Λ
Tan(A+B)
Tan (A + B) β‘
ππππ΄ =
πΆππ π΅ =
ππππ΄+ππππ΅
1βππππ΄ππππ΅
π΄ππ
π»π¦π
πΆππ π΅ = β
12
13
13
ππππ΅ =
πππ
π΄ππ
ππππ΅ =
5
12
5
B
12
5
12
Use Pythagorasβ to find the missing side (ignore negatives)
ππππ΅ = β
90
180
270
360
y = TanΞΈ
Tan is negative in the range 90Λ - 180Λ
7A
Further Trigonometric Identities and their
Applications
You need to know and be able to
use the addition formulae
ππππ΄+ππππ΅
Given that:
SinA =
3
β
5
CosB = β
12
13
Tan (A + B) β‘ 1βππππ΄ππππ΅
180Λ < A < 270Λ
B = Obtuse
Tan (A + B) β‘
3
5
+
β
4
12
3
5
1β 4×β12
Tan (A + B) β‘
1
3
63
48
Find the value of:
Tan(A+B)
3
ππππ΄ =
4
5
ππππ΅ = β
12
1
Work out the
Numerator and
Denominator
Leave, Change and
Flip
ππππ΄+ππππ΅
Tan (A + B) β‘ 1βππππ΄ππππ΅
Substitute in TanA
and TanB
48
Tan (A + B) β‘ 3 × 63
Simplify
16
Tan (A + B) β‘ 63
Although you could just type the whole thing into your
calculator, you still need to show the stages for the
workings marksβ¦
7A
Further Trigonometric Identities and their
Applications
You need to know and be able to use the addition formulae
Given that:
2 π ππ π₯ + π¦ = 3πππ (π₯ β π¦)
Express Tanx in terms of Tanyβ¦
2 π ππ π₯ + π¦ = 3πππ (π₯ β π¦)
2(π πππ₯πππ π¦ + πππ π₯π πππ¦) = 3(πππ π₯πππ π¦ + π πππ₯π πππ¦)
Rewrite the sin and cos parts
Multiply out the brackets
2 π πππ₯πππ π¦ + 2πππ π₯π πππ¦ = 3 πππ π₯πππ π¦ + 3π πππ₯π πππ¦
2 π πππ₯πππ π¦ + 2πππ π₯π πππ¦ = 3 πππ π₯πππ π¦ + 3π πππ₯π πππ¦
πππ π₯πππ π¦
πππ π₯πππ π¦
πππ π₯πππ π¦
πππ π₯πππ π¦
2 π‘πππ₯ + 2π‘πππ¦ = 3 +3π‘πππ₯π‘πππ¦
2 π‘πππ₯ β 3π‘πππ₯π‘πππ¦ = 3 β2π‘πππ¦
π‘πππ₯(2 β 3π‘πππ¦) = 3 β2π‘πππ¦
π‘πππ₯ = 3 β2π‘πππ¦
2 β3π‘πππ¦
Divide all by cosxcosy
Simplify
Subtract 3tanxtany
Subtract 2tany
Factorise the left side
Divide by (2 β 3tany)
7A
Further Trigonometric Identities and their
Applications
You can express sin2A, cos 2A and
tan2A in terms of angle A, using
the double angle formulae
Sin(A + B) β‘ SinACosB + CosASinB
Replace B with A
Sin(A + A) β‘ SinACosA + CosASinA
Simplify
Sin2A β‘ 2SinACosA
1
Sin2A
2
β‘ SinACosA
Sin4A β‘ 2Sin2ACos2A
÷ 2
2A ο 4A
Sin2A β‘ 2SinACosA
x 3
3Sin2A β‘ 6SinACosA
2A = 60
Sin60 β‘ 2Sin30Cos30
7B
Further Trigonometric Identities and their
Applications
You can express sin2A, cos 2A and
tan2A in terms of angle A, using
the double angle formulae
Cos(A + B) β‘ CosACosB - SinASinB
Cos(A + A) β‘ CosACosA - SinASinA
Replace B with A
Simplify
Cos2A β‘ Coπ 2 π΄ β πππ2 π΄
Cos2A β‘ Coπ 2 π΄ β πππ2 π΄
Replace Cos2A with (1 β Sin2A)
Replace Sin2A with (1 β Cos2A)
Cos2A β‘ (1βπππ2 π΄) β πππ2 π΄
Cos2A β‘ Coπ 2 π΄ β (1 - Coπ 2 π΄)
Cos2A β‘ 1 β 2πππ2 π΄
Cos2A β‘ 2Coπ 2 π΄ β 1
7B
Further Trigonometric Identities and their
Applications
You can express sin2A, cos 2A and
tan2A in terms of angle A, using
the double angle formulae
ππππ΄+ππππ΅
Tan (A + B) β‘ 1βππππ΄ππππ΅
Replace B with A
ππππ΄+ππππ΄
Tan (A + A) β‘ 1βππππ΄ππππ΄
Simplify
2ππππ΄
Tan 2A β‘ 1βπππ2π΄
1
Tan 2A
2
β‘
2πππ30
ππππ΄
1βπππ2 π΄
÷ 2
2A = 60
Tan 60 β‘ 1βπππ2 30
2ππππ΄
x 2
2Tan 2A β‘
4ππππ΄
1βπππ2 π΄
Tan 2A β‘ 1βπππ2π΄
2A = A
π΄
Tan A β‘
2πππ 2
π΄
1βπππ2 2
7B
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:
π
π
2π ππ πππ πππ π
2
2
πππ2π β‘ 2π ππππππ π
π
π
ππππ β‘ 2π ππ πππ
2
2
π
π
2π ππ πππ πππ π
2
2
2ΞΈ ο ΞΈ
Replace the first
part
= π ππππππ π
Rewrite
1
= π ππ2π
2
7B
Further Trigonometric Identities and their
Applications
You can express sin2A, cos 2A and
tan2A in terms of angle A, using
the double angle formulae
πΆππ 2π β‘ 2πππ 2 π β 1
πΆππ 4π β‘ 2πππ 2 2π β 1
Double the
angle parts
Show that:
1 + πππ 4π
Can be written as:
2πππ 2 2π
1 + πππ 4π
= 1 + (2πππ 2 2π β 1)
= 2πππ 2 2π
Replace
cos4ΞΈ
The 1s
cancel out
7B
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:
3
πππ π₯ =
4
πΆππ π₯ =
π΄ππ
π»π¦π
4
180Λ < π₯ < 360Λ
ππππ₯ =
7
4
7
3
πΆππ π₯ =
4
x
Use Pythagorasβ to find the missing side (ignore negatives)
Cosx is positive so in the range 270 - 360
π ππ2π₯
Therefore, Sinx is negative
180
πππ
π»π¦π
3
Find the exact value of:
90
ππππ₯ =
270
360
y = CosΞΈ
7
4
Sin2x β‘ 2SinxCosx
Sin2x = 2 ×
y = SinΞΈ
ππππ₯ = β
Sin2x = β
3
×
4
3 7
8
β
7
4
Sub in Sinx and Cosx
Work out and leave in
surd form
7B
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:
3
πππ π₯ =
4
πΆππ π₯ =
π΄ππ
π»π¦π
4
180Λ < π₯ < 360Λ
x
270
360
y = CosΞΈ
270
360
y = TanΞΈ
ππππ₯ = β
7
3
2ππππ₯
Tan 2x β‘ 1βπππ2π₯
Tan 2x =
180
7
3
Use Pythagorasβ to find the missing side (ignore negatives)
Therefore, Tanx is negative
90
ππππ₯ =
7
3
πΆππ π₯ =
4
Cosx is positive so in the range 270 - 360
π‘ππ2π₯
180
πππ
π΄ππ
3
Find the exact value of:
90
ππππ₯ =
2×β
7
Sub in Tanx
7
3
7
1β β 3 ×β 3
πππ2π₯ = β3 7
Work out and leave in
surd form
7B
Further Trigonometric Identities and their
Applications
The double angle formulae allow you
to solve more equations and prove
more identities
Prove the identity:
π‘ππ2π β‘
2
πππ‘π β π‘πππ
π‘ππ2π β‘
2π‘πππ
1 β π‘ππ2 π
2π‘πππ
π‘πππ
π‘ππ2π β‘
1
π‘ππ2 π
β
π‘πππ π‘πππ
π‘ππ2π β‘
Divide each part by
tanΞΈ
Rewrite each part
2
πππ‘π β π‘πππ
7C
Further Trigonometric Identities and their
Applications
The double angle formulae allow you π π π π΄ + π΅ β‘ π πππ΄πππ π΅ + πππ π΄π πππ΅
to solve more equations and prove
more identities
Replace A and B
π π π 2π΄ + π΄ β‘ π ππ2π΄πππ π΄ + πππ 2π΄π πππ΄
By expanding:
π π π 3π΄ β‘ (2π πππ΄πππ π΄)πππ π΄ + (1 β 2π ππ2 π΄)π πππ΄
π ππ(2π΄ + π΄)
Show that:
π π π 3π΄ β‘ 3π πππ΄ β 4π ππ3 π΄
π π π 3π΄ β‘ 2π πππ΄πππ 2 π΄ + π πππ΄ β 2π ππ3 π΄
π π π 3π΄ β‘ 2π πππ΄(1 β π ππ2 π΄) + π πππ΄ β 2π ππ3 π΄
Replace
Sin2A and
Cos 2A
Multiply
out
Replace
cos2A
Multiply out
π π π 3π΄ β‘ 2π πππ΄ β 2π ππ3 π΄ + π πππ΄ β 2π ππ3 π΄
π π π 3π΄ β‘ 3π πππ΄ β 4π ππ3 π΄
Group like
terms
7C
Further Trigonometric Identities and their
Applications
The double angle formulae allow you
to solve more equations and prove
more identities
Given that:
π₯ = 3π πππ and π¦ = 3 β 4πππ 2π
Eliminate ΞΈ and express y in terms of xβ¦
π₯ = 3π πππ
π₯
= π πππ
3
3βπ¦
= πππ 2π
4
3βπ¦
π₯
= 1β2
4
3
2
Replace Cos2ΞΈ and
SinΞΈ
Multiply by 4
π₯
3βπ¦ = 4β8
3
2
Subtract 3
βπ¦ = 1 β 8
Divide
by 3
π¦ = 3 β 4πππ 2π
πππ 2π = 1 β 2π ππ2 π
π₯
3
2
Multiply by -1
π¦ = 8
π₯
3
2
β1
Subtract 3, divide by 4
Multiply by -1
7C
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:
3πππ 2π₯ β πππ π₯ + 2 = 0
3πππ 2π₯ β πππ π₯ + 2 = 0
3(2πππ 2 π₯ β 1) β πππ π₯ + 2 = 0
y = CosΞΈ
2
3
90
180
270
Group terms
2
6πππ π₯ β πππ π₯ β 1 = 0
Factorise
(3πππ π₯ + 1)(2πππ π₯ β 1) = 0
(All trigonometrical parts must be in terms
x, rather than 2x)
β1
Multiply out the
bracket
6πππ 2 π₯ β 3 β πππ π₯ + 2 = 0
0° β€ π₯ β€ 360°
1
Replace cos2x
360
πππ π₯ = β
1
1
or πππ π₯ =
3
2
Solve both pairs
π₯ = πππ β1 β
1
3
π₯ = 109.5° , 250.5°
π₯ = πππ β1
1
2
Remember to find
additional answers!
π₯ = 60° , 300°
π₯ = 60°, 109.5°, 250.5°, 300°
7C
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:
3π πππ₯ + 4πππ π₯
Can be expressed in the form:
π
π ππ(π₯ + Ξ±)
π
>0
0° < Ξ± < 90°
So:
3π πππ₯ + 4πππ π₯
= 5sin(π₯ + 53.1°)
π
π ππ(π₯ + Ξ±) = π
π πππ₯πππ Ξ± + π
πππ π₯π ππΞ±
3π πππ₯ + 4πππ π₯ = π
π πππ₯πππ Ξ± + π
πππ π₯π ππΞ±
π
πππ Ξ± = 3
πππ Ξ± =
π
π ππΞ± = 4
π΄
π»
3
π
π ππΞ± =
4
π
π
π»
So in the triangle, the Hypotenuse is Rβ¦
π
=
32 + 42
πππ Ξ± =
πππ Ξ± =
3
π
Ξ± = 53.1°
Compare each term β
they must be equal!
π
4
Ξ±
3
π
=5
R=5
3
5
Ξ± = πππ β1
Replace with the
expression
3
5
Inverse Cos
Find the smallest value in the
acceptable range given
7D
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
π
π ππ(π₯ β Ξ±) = π
π πππ₯πππ πΌ β π
πππ π₯π πππΌ
π πππ₯ β 3πππ π₯ = π
π πππ₯πππ πΌ β π
πππ π₯π πππΌ
π
π πππΌ = 3
π
πππ πΌ = 1
Show that you can express:
π πππ₯ β 3πππ π₯
In the form:
π
>0
π
π ππ(π₯ β Ξ±)
0<Ξ±<
π
2
π
=
12 +
3
π
πππ πΌ = 1
πππ πΌ =
π πππ₯ β 3πππ π₯
= 2sin π₯ β
π
3
πΌ=
π
3
π
=2
Divide
by 2
1
2
πΌ = πππ β1
Compare each term β
they must be equal!
R=2
2πππ πΌ = 1
So:
2
Replace with the
expression
1
2
Inverse
cos
Find the smallest
value in the
acceptable range
7D
Further Trigonometric Identities and their
Applications
Sketch the graph of: π πππ₯ β 3πππ π₯
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:
π πππ₯ β 3πππ π₯
In the form:
π
π ππ(π₯ β Ξ±)
π
>0
0<Ξ±<
π
2
So:
π πππ₯ β 3πππ π₯
= 2sin π₯ β
π
2sin β
3
=β 3
= Sketch the graph of: 2sin π₯ β
1
π
3
Ο/
2
Ο
3Ο/
2
1
Ο/
3
Ο/
2
Ο
4Ο/
3
3Ο/
2
π
3
2Ο
y = 2sin π₯ β
1
-1
-2
2Ο
y = sin π₯ β
2
At the yintercept,
x=0
Start out
with sinx
y = sin π₯
-1
-1
π
3
Ο/
3
Ο/
2
Ο
4Ο/
3
3Ο/
2
2Ο
π
3
Translate
Ο/3 units
right
Vertical
stretch, scale
factor 2
7D
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:
2πππ π + 5π πππ
in the form:
π
πππ (π β πΌ)
π
>0
0° < πΌ < 90°
So:
2πππ π + 5π πππ
= 29cos(π β 68.2)
π
πππ (π β πΌ) = π
πππ ππππ πΌ + π
π ππππ πππΌ
2πππ π + 5π πππ = π
πππ ππππ πΌ + π
π ππππ πππΌ
π
πππ πΌ = 2
π
=
π
π πππΌ = 5
22 +52
π
πππ πΌ = 2
29πππ πΌ = 2
πππ πΌ =
2
29
πΌ = πππ β1
πΌ = 68.2
2
29
Replace with the
expression
Compare each term β
they must be equal!
π
= 29
R = β29
Divide by
β29
Inverse
cos
Find the smallest
value in the
acceptable range
7D
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
29cos(π β 68.2) = 3
Divide by β29
cos(π β 68.2) =
3
29
Inverse Cos
Solve in the given range, the
following equation:
3
π β 68.2 = πππ β1
29
2πππ π + 5π πππ = 3
π β 68.2 = 56.1, β56.1 , 303.9
0° < π < 360°
We just showed that the original equation can
be rewrittenβ¦
2πππ π + 5π πππ = 29cos(π β 68.2)
Remember to work out
other values in the
adjusted range
Add 68.2 (and
put in order!)
π = 12.1 , 124.3
Hence, we can solve this equation instead!
29cos(π β 68.2) = 3
0° < π < 360°
β68.2° < π β 68.2 < 291.2°
-56.1
Remember to
adjust the
range for (ΞΈ β
68.2)
-90
56.1
90
303.9
180
270
y = CosΞΈ
360
7D
Further Trigonometric Identities and their
Applications
Rcos(ΞΈ β Ξ±) chosen as it
gives us the same form
as the expression
You can write expressions of the
form acosΞΈ + bsinΞΈ, where a and
b are constants, as a sine or
cosine function only
π
πππ (π β πΌ) = π
πππ ππππ πΌ + π
π ππππ πππΌ
12πππ π + 5π πππ = π
πππ ππππ πΌ + π
π ππππ πππΌ
π
πππ πΌ = 12
Find the maximum value of the
following expression, and the
smallest positive value of ΞΈ at which
it arises:
12πππ π + 5π πππ
= 13cos(π β 22.6)
13cos(π β 22.6)
13(1)
πππ₯ = 13
π β 22.6 = 0
π = 22.6
Max value of
cos(ΞΈ - 22.6) = 1
Overall maximum
therefore = 13
Cos peaks at 0
ΞΈ = 22.6 gives us 0
π
=
π
π πππΌ = 5
122 +52
π
πππ πΌ = 12
13πππ πΌ = 12
12
πππ πΌ =
13
πΌ = πππ β1
πΌ = 22.6
12
13
Replace with the
expression
Compare each term β
they must be equal!
π
= 13
R = 13
Divide by 13
Inverse
cos
Find the smallest
value in the
acceptable range
7D
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
ππ πππ ± ππππ π
π
π ππ π ± πΌ
ππππ π ± ππ πππ
π
πππ π β πΌ
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!
7D
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
π πππ + π πππ = 2π ππ
π+π
πβπ
πππ
2
2
π+π
πβπ
π πππ β π πππ = 2πππ
π ππ
2
2
πππ π + πππ π = 2πππ
π+π
πβπ
πππ
2
2
πππ π β πππ π = β2π ππ
π+π
πβπ
π ππ
2
2
You get given all these in the
formula booklet!
7E
Further Trigonometric Identities and their
Applications
Using the formulae for Sin(A + B) and Sin (A β B),
derive the result that:
You can express sums and differences of
sines and cosines as products of sines
and cosines by using the βfactor
formulaeβ
π+π
πβπ
π πππ + π πππ = 2π ππ
πππ
2
2
π πππ β π πππ = 2πππ
2) πππ π΄ β π΅ = π πππ΄πππ π΅ β πππ π΄π πππ΅
πππ π΄ + π΅ + π ππ(π΄ β π΅) = 2π πππ΄πππ π΅
πππ π΄ + π΅ + π ππ(π΄ β π΅) = 2π πππ΄πππ π΅
π+π
πβπ
ππππ + ππππ = 2π ππ
πππ
2
2
Add both sides
together (1 + 2)
Let (A+B) = P
Let (A-B) = Q
π+π
πβπ
πππ π + πππ π = 2πππ
πππ
2
2
π+π
πβπ
π ππ
2
2
π+π
πβπ
πππ
2
2
1) πππ π΄ + π΅ = π πππ΄πππ π΅ + πππ π΄π πππ΅
π+π
πβπ
π ππ
2
2
πππ π β πππ π = β2π ππ
π πππ + π πππ = 2π ππ
1)
π΄+π΅ =π
2)
π΄βπ΅ =π
2π΄ = π + π
π΄=
π+π
2
1+2
Divide
by 2
1)
π΄+π΅ =π
2)
π΄βπ΅ =π
2π΅ = π β π
πβπ
π΅=
2
1-2
Divide
by 2
7E
Further Trigonometric Identities and their
Applications
Show that:
You can express sums and differences of
sines and cosines as products of sines
and cosines by using the βfactor
formulaeβ
π πππ + π πππ = 2π ππ
π πππ β π πππ = 2πππ
πππ π + πππ π = 2πππ
π+π
πβπ
πππ
2
2
π+π
πβπ
π ππ
2
2
π+π
πβπ
πππ
2
2
π+π
πβπ
πππ π β πππ π = β2π ππ
π ππ
2
2
π πππ β π πππ = 2πππ
π ππ105 β π ππ15 =
π+π
πβπ
π ππ
2
2
π ππ105 β π ππ15 = 2πππ
105 + 15
105 β 15
π ππ
2
2
π ππ105 β π ππ15 = 2πππ 60π ππ45
π ππ105 β π ππ15 = 2 ×
π ππ105 β π ππ15 =
1
1
1
1
×
2
2
2
P = 105
Q = 15
Work out the
fraction parts
Sub in values for
Cos60 and Sin45
Work out the
right hand side
2
7E
Further Trigonometric Identities and their
Applications
Solve in the range indicated:
You can express sums and differences of
π ππ4π β π ππ3π = 0
0β€πβ€π
sines and cosines as products of sines
and cosines by using the βfactor
π+π
πβπ
π πππ β π πππ = 2πππ
π ππ
formulaeβ
2
π+π
πβπ
π πππ + π πππ = 2π ππ
πππ
2
2
π ππ4π β π ππ3π = 2πππ
4π + 3π
4π β 3π
π ππ
2
2
π ππ4π β π ππ3π = 2πππ
7π
π
π ππ
2
2
π+π
πβπ
π πππ β π πππ = 2πππ
π ππ
2
2
πππ π + πππ π = 2πππ
π+π
πβπ
πππ
2
2
2πππ
π+π
πβπ
πππ π β πππ π = β2π ππ
π ππ
2
2
0β€πβ€π
πππ
Adjust the
range
7π 7π
0β€
β€
2
2
Ο/
2
Ο
3Ο/
2
7π
π
π ππ
=0
2
2
7π
=0
2
7π
= πππ β1 0
2
0
2Ο
y = CosΞΈ
2
7π π 3π 5π 7π
,
= ,
,
2
2 2 2 2
π 3π 5π
π= ,
,
,π
7 7 7
P = 4ΞΈ
Q = 3ΞΈ
Work 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
7E
Further Trigonometric Identities and their
Applications
Solve in the range indicated:
You can express sums and differences of
π ππ4π β π ππ3π = 0
0β€πβ€π
sines and cosines as products of sines
and cosines by using the βfactor
π+π
πβπ
π πππ β π πππ = 2πππ
π ππ
formulaeβ
2
π+π
πβπ
π πππ + π πππ = 2π ππ
πππ
2
2
π ππ4π β π ππ3π = 2πππ
4π + 3π
4π β 3π
π ππ
2
2
π ππ4π β π ππ3π = 2πππ
7π
π
π ππ
2
2
π+π
πβπ
π πππ β π πππ = 2πππ
π ππ
2
2
πππ π + πππ π = 2πππ
π+π
πβπ
πππ
2
2
0β€πβ€π
0
Ο/
2
π ππ
Adjust the
range
π π
0β€ β€
2 2
Ο
3Ο/
2
7π
π
π ππ
=0
2
2
2πππ
π+π
πβπ
πππ π β πππ π = β2π ππ
π ππ
2
2
π
=0
2
π
= π ππβ1 0
2
2Ο
y = SinΞΈ
2
π
=0
2
π=0
P = 4ΞΈ
Q = 3ΞΈ
Work 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
7E
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
π πππ + π πππ = 2π ππ
π+π
πβπ
πππ
2
2
π πππ β π πππ = 2πππ
π+π
πβπ
π ππ
2
2
π+π
πβπ
πππ π + πππ π = 2πππ
πππ
2
2
πππ π β πππ π = β2π ππ
π+π
πβπ
π ππ
2
2
Prove that:
π ππ π₯ + 2π¦ + π ππ π₯ + π¦ + π πππ₯
= π‘ππ(π₯ + π¦)
πππ π₯ + 2π¦ + πππ π₯ + π¦ + πππ π₯
Numerator:
In the numerator:
π ππ π₯ + 2π¦ + π ππ π₯ + π¦ + π πππ₯
π ππ π₯ + 2π¦ + π πππ₯
Ignore sin(x + y)
for nowβ¦
π+π
πβπ
π πππ + π πππ = 2π ππ
πππ
2
2
π ππ(π₯ + 2π¦) + π πππ₯ = 2π ππ
Use the identity
for adding 2 sines
π₯ + 2π¦ + π₯
π₯ + 2π¦ β π₯
πππ
2
2
= 2 π ππ π₯ + π¦ πππ π¦
= 2 π ππ π₯ + π¦ πππ π¦ + π ππ(π₯ + π¦)
P = x + 2y
Q=x
Simplify
Fractions
Bring back the sin(x + y)
we ignored earlier
Factorise
= π ππ π₯ + π¦ (2πππ π¦ + 1)
π ππ π₯ + π¦ (2πππ π¦ + 1)
7E
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
π πππ + π πππ = 2π ππ
π+π
πβπ
πππ
2
2
π πππ β π πππ = 2πππ
π+π
πβπ
π ππ
2
2
π+π
πβπ
πππ π + πππ π = 2πππ
πππ
2
2
πππ π β πππ π = β2π ππ
π+π
πβπ
π ππ
2
2
Prove that:
π ππ π₯ + 2π¦ + π ππ π₯ + π¦ + π πππ₯
= π‘ππ(π₯ + π¦)
πππ π₯ + 2π¦ + πππ π₯ + π¦ + πππ π₯
Numerator:
In the denominator:
πππ π₯ + 2π¦ + πππ π₯ + π¦ + πππ π₯
πππ π₯ + 2π¦ + πππ π₯
Ignore cos(x + y)
for nowβ¦
π+π
πβπ
πππ π + πππ π = 2πππ
πππ
2
2
πππ (π₯ + 2π¦) + πππ π₯ = 2πππ
Use the identity for
adding 2 cosines
π₯ + 2π¦ + π₯
π₯ + 2π¦ β π₯
πππ
2
2
= 2 πππ π₯ + π¦ πππ π¦
= 2 πππ π₯ + π¦ πππ π¦ + πππ (π₯ + π¦)
P = x + 2y
Q=x
Simplify
Fractions
Bring back the cos(x + y)
we ignored earlier
Factorise
= πππ π₯ + π¦ (2πππ π¦ + 1)
π ππ π₯ + π¦ (2πππ π¦ + 1)
Denominator: πππ π₯ + π¦ (2πππ π¦ + 1)
7E
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
π πππ + π πππ = 2π ππ
π+π
πβπ
πππ
2
2
π πππ β π πππ = 2πππ
π+π
πβπ
π ππ
2
2
π+π
πβπ
πππ π + πππ π = 2πππ
πππ
2
2
πππ π β πππ π = β2π ππ
π ππ π₯ + 2π¦ + π ππ π₯ + π¦ + π πππ₯
πππ π₯ + 2π¦ + πππ π₯ + π¦ + πππ π₯
=
=
π ππ(π₯ + π¦)(2πππ π¦ + 1)
πππ (π₯ + π¦)(2πππ π¦ + 1)
π ππ(π₯ + π¦)
πππ (π₯ + π¦)
Replace the
numerator and
denominator
Cancel out the
(2cosy + 1) brackets
Use one of the
identities from C2
= π‘ππ(π₯ + π¦)
π+π
πβπ
π ππ
2
2
Prove that:
π ππ π₯ + 2π¦ + π ππ π₯ + π¦ + π πππ₯
= π‘ππ(π₯ + π¦)
πππ π₯ + 2π¦ + πππ π₯ + π¦ + πππ π₯
Numerator:
π ππ π₯ + π¦ (2πππ π¦ + 1)
Denominator: πππ π₯ + π¦ (2πππ π¦ + 1)
7E
Summary
β’ 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