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Basic Trigonometric Identities
2Higher Maths 2 3 Advanced Trigonometry
There are several basic trigonometric factsor identities which it is important to remember.
( sin x ) 2
is written
sin 2
x
sin 2
x + cos 2
x = 1
cos 2
x = 1 – sin 2
x
tan x =sin xcos x
Alternatively,
Example Find tan x if sin x
=cos
2 x = 1 – sin
2 x
= 1 –
= 59
cos x =
tan x =2
sin 2
x = 1 – cos 2
x 23 ÷ =3
55
35
49
23
Compound Angles
3Higher Maths 2 3 Advanced Trigonometry
An angle which is the sum of two other angles is called a Compound Angle.
Angle SymbolsGreek letters are often used for angles.
‘Alpha
’‘Beta’
‘Theta
’‘Phi’
‘Lambd
a’
B
C
A
BAC =
BAC is acompound angle.
sin ( ) sin + sin
+
+ ≠
sin ( )Formula for
4
By extensive working,it is possible to prove that
+
sin ( )+ sin +sin≠
sin ( ) = sin cos + sin cos
+
Higher Maths 2 3 Advanced Trigonometry
Example
Find the exact value of sin 75°
sin 75° = sin ( 45° + 30° )
= sin 45°cos 30° + sin 30°cos 45°
= 2
312
× + × = 2 2+1
2132
1
Compound Angle Formulae
5
sin ( ) = sin cos + sin
cos +
Higher Maths 2 3 Advanced Trigonometry
sin ( ) = sin cos – sin
cos –
cos ( ) = cos cos – sin
sin+
cos ( ) = cos cos + sin
sin–
The result for sin ( )+ can be used to findall four basic compound angle
formulae.
Proving Trigonometric Identities
6Higher Maths 2 3 Advanced Trigonometry
Example
Prove the identity
sin ( )+
cos costan + tan=
sin ( )+
cos cos=
cos cos
sin cos +sin cos
=cos cos
sin cos+
sin cos
cos cos
=cos
sin+
sin
cos= tan + tan
An algebraic fact is called an identity.
tan x sin xcos x=
‘Left Hand Side’
L.H.S.
R.H.S. ‘Right
Hand Side’
Applications of Trigonometric Addition Formulae
7Higher Maths 2 3 Advanced Trigonometry
K
L
J
M
8
3 4
From the diagram, show
thatcos ( )– =
255
KL
= 82 + 42
80
= = 4 5
JK
= 32 + 42
25
= = 5Example
cos ( )– = cos cos + sin sin
=5
15
× + ×34 2
5 5
10 55
= = 25
255
=
cos =454
15
= sin =854
25
=
Find any unknown sides:
Investigating Double Angles
8Higher Maths 2 3 Advanced Trigonometry
The sum of two identical angles can be written as and is called a double angle.
2
2sin = sin ( + ) = cos +sin cossin
= 2sin cos
2cos = cos ( + ) = cos –cos sinsin
= cos 2 – sin
2
= cos 2 – ( 1 – cos
2 )
= cos 2 – 12 or sin
2–1 2
sin 2
x + cos 2
x = 1
sin 2
x = 1 – cos 2
x
( )
Double Angle Formulae
9Higher Maths 2 3 Advanced Trigonometry
There are several basic
identities for double
angles which it is
useful to know.
sin 2 = 2 sin cos
cos 2 = cos2 – sin2
= 2 cos2 – 1
= 1 – 2 sin2Example
3
4
If tan = ,
calculate
and .
34
5
sin 2 cos 2
sin 2 =2 sin cos
= 2 × 54 × 5
3
= 25
24
cos 2 = cos 2 – sin
2
= 53 –
= 25
7
2 ( )54 2
–
tan = adjopp
Trigonometric Equations involving Double Angles
10
Higher Maths 2 3 Advanced Trigonometry
cos 2 x – cos x = 0 Solve
for
0 x 2π
cos 2 x – cos x = 0
2 cos 2
x – 1 – cos x = 0 2 cos
2 x – cos x – 1 =
0 ( 2 cos x + 1) ( cos x – 1) = 0
cos x – 1 = 0
cos x = 1
x = 2π
2 cos x + 1 = 0
cos x = 21–S A
T
3π
x =C 3π4
3π2
or x = x = 0
or
or
substitu
te
remember
Example
Intersection of Trigonometric Graphs
11
Higher Maths 2 3 Advanced Trigonometry
4
-4
360°
A
B
f (x)g(x)
Example
The diagram opposite shows
the graphs of and
.
g(x)f (x)
Find the x-coordinate of A and B.
4 sin 2 x = 2 sin x
4 sin 2 x – 2 sin x = 0
4 × ( 2 sin x cos x ) – 2 sin x = 0
8 sin x cos x – 2 sin x = 0
2 sin x ( 4 cos x – 1 ) = 0 common factor
f (x) =
g(x) 2 sin x = 0
4 cos x – 1 = 0 or
x = 0°, 180° or 360°
orx ≈ 75.5° or 284.5°
Solving by trigonometry,
Quadratic Angle Formulae
12
Higher Maths 2 3 Advanced Trigonometry
The double angle formulae
can also be rearranged to
give quadratic angle
formulae.
cos2 = 21 ( 1 + cos 2
)sin2 = 2
1 ( 1 – cos 2
)Example
Express
in terms of cos 2 x .
2 cos 2
x – 3 sin 2
x2 × ( 1 + cos 2 x ) – 3 × ( 1 – cos
2 x ) 21
21
1 + cos 2 x – +
cos 2 x
23
23
25
21– + cos 2 x
=
=
= = 21( 5 cos 2 x – 1 )
substitute
Quadratic means
‘squared’
2 cos 2
x – 3 sin 2
x
Angles in Three Dimensions
13
Higher Maths 2 3 Advanced Trigonometry
In three dimensions, a flat
surface is called a plane.
Two planes at different
orientations have a straight
line of intersection.
A
B
C
D
P
Q
J
LK The angle between two
planes is defined as
perpendicular to the line of
intersection.