8/4/2019 H2 Mathematics - Trigonometry
http://slidepdf.com/reader/full/h2-mathematics-trigonometry 1/12
Page 1 of 12
P
QO
H2 Mathematics – Essentials for Trigonometry
1 Objective
This set of materials has been developed for students to acquire just-in-time skills in
trigonometry (which is not a formal topic in the syllabus), which may be applied in other
areas of H2 Mathematics, e.g. summation of series, differentiation, integration, complexnumbers, etc.
You should spend at least 3 hours in understanding the concepts & applying them.
Reference books:
Pure Mathematics by L Bostok & S Chandler
New Additional Mathematics by Ho Soo Thong & Khor Nyak Hiong
2 Basic concepts
2.1 Trigonometric ratios
adjacent sidecos
hypothenuse
OQ
OPθ = =
opposite sidesin
hypothenuse
PQ
OPθ = =
opposite sidetan
adjacent side
PQ
OQθ = =
2.2 Signs of trigonometric ratios in the four quadrants
2nd quadrant
only sinθ positive (S)
1st quadrant
all 3 ratios positive (A)
3rd quadrant
only tanθ positive (T)
4th quadrant
only cosθ positive (C)
θ
8/4/2019 H2 Mathematics - Trigonometry
http://slidepdf.com/reader/full/h2-mathematics-trigonometry 2/12
Page 2 of 12
2.3 Trigonometric identities
sintan
cos
θ θ
θ =
1cot
tan
θ θ
=
1sec
cosθ
θ =
1cosec
sinθ
θ =
Note: denominator in
each identity cannot be
zero.
Pythagorean identity: 2 2sin cos 1θ θ + = -- (*)
Dividing (*) throughout by 2sin θ , we obtain2 21 cot cosecθ θ + =
Dividing (*) throughout by2
cos θ , we obtain2 2tan 1 secθ θ + =
2.4 Useful relationships
Negative angles: ( )cos cosθ θ − =
( )sin sinθ θ − = −
( )tan tanθ θ − = −
Complementary angles: cos sin2
π θ θ
− =
sin cos2
π θ θ
− =
tan cot2
π
θ θ
− =
cot tan2
π θ θ
− =
For example,1
sin sin6 6 2
π π − = − = −
Note: π radians = 180
8/4/2019 H2 Mathematics - Trigonometry
http://slidepdf.com/reader/full/h2-mathematics-trigonometry 3/12
Page 3 of 12
2.5 Special angles
θ 0 6
π
4
π
3
π
2
π
π 3
2
π
2π
sinθ 0 1
2
1
2
3
2
1 0 1− 0
cosθ 1 3
2
1
2
1
2
0 1− 0 1
tanθ 0 1
3
1 3 undefined 0 undefined 0
3 Compound angle identities
For any 2 angles A and B, A B+ , A B− and 2 A B+ are called compound angles.
We can prove that: ( )sin sin cos cos sin x y x y x y+ = + -- (1) and
( )cos cos cos sin sin x y x y x y+ = − -- (2)
If you are interested, you may wish to refer to the geometry proofs of (1) and (2) at:
http://www.acts.tinet.ie/compoundanglesandcalcu_668.html
From (1), we replace y with y− to obtain ( )( ) ( )sin sin x y x y+ − = −
( ) ( )sin cos cos sin x y x y= − + −
sin cos cos sin x y x y= − -- (3)
From (2), we use the same approach to obtain ( )( ) ( )cos cos x y x y+ − = −
cos cos sin sin x y x y= + -- (4)
Consider (1) ÷ (2): ( )sin cos cos sin
tancos cos sin sin
x y x y x y
x y x y
++ =
−
Dividing each term in the numerator and denominator by cos cos x y ,
( )
sin cos cos sincos cos
tancos cos sin sin
cos cos
x y x y x y
x y x y x y
x y
+
+ =−
tan tan
1 tan tan
x y
x y
+=
−-- (5)
8/4/2019 H2 Mathematics - Trigonometry
http://slidepdf.com/reader/full/h2-mathematics-trigonometry 4/12
Page 4 of 12
From (5), we replace y with y− to obtain ( )( ) ( )tan tan x y x y+ − = −
tan tan
1 tan tan
x y
x y
−=
+-- (6)
Worked example 1: Without using a calculator, show that3 1
cos752 2
° −= . Similarly,
find (a) cos15° , (b) tan105° , (c) ( )sin 15°− .
( )cos 75 cos 30 45° ° °= + [ Note: 30
°& 45
°are special angles ]
cos 30 cos 45 sin 30 sin 45° ° ° °
= − [ Applying formula (2) ]
3 1 1 1
2 22 2= × − ×
3 12 2
−= (shown)
(a) ( )cos15 cos 45 30° ° °= −
cos 45 cos 30 sin 45 sin 30° ° ° °
= + [ Applying formula (4) ]
1 3 1 1
2 22 2= × + ×
3 1
2 2
+=
(b) ( )tan105 tan 60 45° ° °
= +
tan 60 tan 45
1 tan 60 tan 45
° °
° °
+=
−[ Applying formula (5) ]
3 1
1 3 1
+=
− ×
3 1
1 3
+=
−
8/4/2019 H2 Mathematics - Trigonometry
http://slidepdf.com/reader/full/h2-mathematics-trigonometry 5/12
Page 5 of 12
(c) ( ) ( )sin 15 sin 30 45° ° °− = − [ Applying formula (3) ]
sin 30 cos 45 cos 30 sin 45° ° ° °
= −
1 1 3 1
2 22 2= × − ×
1 32 2−=
4 Double angle identities
We make use of (1), (2), (5) in Section 3 to obtain double angle formulae sin 2 x , cos2 x ,
tan 2 x respectively.
From (1), we replace y with x to obtain ( )sin sin 2 x x x+ =
sin cos cos sin x x x x= +
2 sin cos x x= -- (7)
From (2), we use the same approach to obtain ( )cos cos 2 x x x+ =
cos cos sin sin x x x x= − 2 2cos sin x x= −
21 2sin x= − 2
2cos 1 x= − -- (8)
From (5), we use the same approach to obtain ( )tan tan 2 x x x+ =
tan tan1 tan tan
x x x x+=
−
2
2tan
1 tan
x
x=
−-- (9)
8/4/2019 H2 Mathematics - Trigonometry
http://slidepdf.com/reader/full/h2-mathematics-trigonometry 6/12
Page 6 of 12
Worked example 2: Without using a calculator, find the values of sin 2 A , cos2 A and
tan 2 A if (a)3
sin5
A = and A is acute, (b)1
cos2
A = − and A is obtuse,
(c) cot 2 A = and 90 270 A° °
< < , (d)2
sec3
A = and 90 0 A° °
− < < .
We will use formulae (7), (8), (9) for each part of this example.
(a)3
sin5
A = , so4
cos5
A = and3
tan4
A = from the diagram
3 4 24sin 2 2sin cos 2
5 5 25 A A A∴ = = × × =
2 2
2 2 4 3 7cos 2 cos sin
5 5 25 A A A
∴ = − = − =
22
3 32
2 tan 244 2tan271 tan 73
1164
A A
A
×
∴ = = = =−
−
(b)1
cos2
A = − , so3
sin2
A = and tan 3 A = − from the diagram
3 1 3sin 2 2sin cos 2
2 2 2 A A A∴ = = × × − = −
22
2 2 1 3 1cos 2 cos sin
2 2 2 A A A
∴ = − = − − = −
( )22
2 tan 2 3tan 2 3
1 tan 1 3
A A
A
× −∴ = = =
−− −
A
3
4
5
2
−1
3 A
8/4/2019 H2 Mathematics - Trigonometry
http://slidepdf.com/reader/full/h2-mathematics-trigonometry 7/12
Page 7 of 12
(c)1
cot 2 tan2
A A= ⇒ = . Since tan A is positive, A has to lie in the third quadrant.
So,1
sin5
A = − and2
cos5
A = − from the diagram
1 2 4sin 2 2sin cos 255 5
A A A∴ = = × − × − =
2 2
2 2 2 1 3cos 2 cos sin
55 5 A A A
∴ = − = − − − =
22
12
2 tan 42tan21 tan 31
12
A A
A
×
∴ = = =−
−
(d)2 3
sec cos23
A A= ⇒ = ; since 90 0 A° °
− < < , A is in the fourth quadrant.
So1
sin2
A = − and1
tan3
A = −
(Try drawing the diagram yourself to obtain the ratios)
1 3 3sin 2 2sin cos 2
2 2 2
A A A∴ = = × − × = −
2 2
2 2 3 1 1cos 2 cos sin
2 2 2 A A A
∴ = − = − − =
22
12
2tan 3tan 2 3
1 tan 11
3
A A
A
× −
∴ = = = −−
− −
−2
5−1
A
8/4/2019 H2 Mathematics - Trigonometry
http://slidepdf.com/reader/full/h2-mathematics-trigonometry 8/12
Page 8 of 12
By replacing x with2
xin (7), (8), (9) we can obtain expressions for sin x , cos x , tan x
respectively. These are half-angle identities.
For example, sin 2 sin 2sin cos
2 2 2
x x x x
= =
Similarly, 2 2 2 2cos cos sin 1 2sin 2cos 12 2 2 2 2
x x x x x= − = − = −
and2
2tan2tan
1 tan2
x
x x
=
−
.
By considering ( )sin 2 x x+ and the double angle formula for sine, we can obtain an
expression for sin 3 x in terms of sin x as well as cos3 x in terms of cos x . These aremultiple angle identities.
For example, ( )sin 2 sin 3 x x x+ =
sin cos 2 cos sin 2 x x x x= +
( ) ( )2sin 1 2sin cos 2sin cos x x x x x= − +
3 2sin 2sin 2sin cos x x x x= − +
( )3 2sin 2sin 2sin 1 sin x x x x= − + −
33sin 4sin x x= −
By a similar approach, ( ) 3cos 2 cos3 4cos 3cos x x x x x+ = = −
Try to apply the approach and see for yourself.
8/4/2019 H2 Mathematics - Trigonometry
http://slidepdf.com/reader/full/h2-mathematics-trigonometry 9/12
Page 9 of 12
5 Factor formulae
We make use of (1), (2), (3), (4) in Section 3 to obtain 4 factor formulae (or product-to-
sum identities):
Consider (1) + (3):
( ) ( )sin sin sin cos cos sin sin cos cos sin x y x y x y x y x y x y+ + − = + + −
2 sin cos x y= -- (10)
Consider (1) − (3): ( ) ( )sin sin 2cos sin x y x y x y+ − − = -- (11)
Consider (2) + (4):
( ) ( )cos cos cos cos sin sin cos cos sin sin x y x y x y x y x y x y+ + − = − + +
2 cos cos x y= -- (12)
Consider (2) − (4), ( ) ( )cos cos 2sin sin x y x y x y+ − − = − -- (13)
Identities (10), (11), (12), (13) should be used when a given product is to be changed to a
sum or difference.
For example, to express 2cos7 cos 2θ θ as a sum we use (12) to give
( ) ( )cos 7 2 cos 7 2 cos9 cos5θ θ θ θ θ θ + + − = + .
Suppose in (10), we let x y p+ = and x y q− = . Then
2
p q x
+= and
2
p q y
−= .
Substitute these into the identity to obtain sin sin 2sin cos2 2
p q p q p q
+ −+ = -- (12)
We use the same approach for (11) to derive sin sin 2cos sin2 2
p q p q p q
+ −− = -- (13)
Similarly, cos cos 2cos cos2 2
p q p q p q
+ −+ = -- (14)
and cos cos 2sin sin2 2
p q p q p q+ −
− = − -- (15)
8/4/2019 H2 Mathematics - Trigonometry
http://slidepdf.com/reader/full/h2-mathematics-trigonometry 10/12
Page 10 of 12
Identities (12), (13), (14), (15) are best used when a sum or difference is to be expressed
as a product. We call these sum-to-product identities.
For example, to express sin 6 sin4θ θ − as a product we would use (13) to obtain
6 4 6 42cos sin 2cos5 sin
2 2
θ θ θ θ θ θ
+ −= .
Worked example 3: Prove thatsin sin
tancos cos 2
A B A B
A B
+ +≡
+. If A, B and C are the angles of
a triangle, deduce thatsin sin
cotcos cos 2
A B C
A B
+=
+.
sin sinLHS
cos cos
A B
A B
+≡
+
[ Applying formula (12) for the numerator & formula (14) for the denominator ]
2sin cos2 2
2cos cos2 2
A B A B
A B A B
+ −
≡+ −
tan2
A B+≡
Since A, B and C are angles in a triangle, A B C π + + =
2 2 2
A B C π +⇒ + =
2
A B+ ⇒
and
2
C are complementary
tan tan cot2 2 2 2
A B C C π + ∴ = − =
[ Recall the useful relationships in Section 2.4 ]
8/4/2019 H2 Mathematics - Trigonometry
http://slidepdf.com/reader/full/h2-mathematics-trigonometry 11/12
Page 11 of 12
Worked example 4: Prove cos cos3 cos5 cos 7 4cos 4 sin 2 sinθ θ θ θ θ θ θ − − + ≡ − .
LHS cos cos3 cos5 cos 7θ θ θ θ ≡ − − +
( ) ( )cos7 cos cos5 cos3θ θ θ θ ≡ + − + [ Applying formula (14) twice ]
7 7 5 3 5 32cos cos 2cos cos
2 2 2 2
θ θ θ θ θ θ θ θ + − + −≡ −
2cos 4 cos3 2cos 4 cosθ θ θ θ ≡ −
( )2cos 4 cos3 cosθ θ θ ≡ −
3 32cos 4 2sin sin
2 2
θ θ θ θ θ
+ − ≡ −
[ Applying formula (15) ]
4 cos 4 sin 2 sinθ θ θ ≡ −
6 Self-attempt Exercises
Apply compound angle identities to solve Q1 & Q2.
Q1 Without using a calculator, evaluate:
(a) cos80 cos20 sin80 sin20° ° ° °+ , (b) sin 37 cos 7 cos 37 sin 7° ° ° °
− ,
(c) sin165°, (d) tan75
°.
[ Answers: (a)1
2, (b)
1
2, (c) ( )
16 2
4− , (d) 2 3+ ]
Q2 Prove ( )
( )
sintan tan
cos cos
y x y x
x x y+ − ≡
+.
Apply double angle identities to solve Q3 & Q4.
Q3 Given that cos cθ = and that θ is acute, express in terms of c,
(a) cos2θ , (b) sin2θ , (c) tan 2θ , (d) sin2
θ .
[ Answers: (a) 22 1c − , (b) 22 1c c− , (c)2
2
2 1
2 1
c c
c
−
−, (d)
1
2
c−]
Q4 Provecos sin
sec 2 tan 2cos sin
A A A A
A A
++ ≡
−.
Top Related