MATHEMATICS EXTENSION 2
4 UNIT MATHEMATICS
GEOMETRY and TRIGONOMETRY
Essential Background Topic
The topics of geometry and trigonometry are essential in the study of most of
mathematics and is a fundamental topic in mathematics, physics, chemistry,
engineering etc.
1. CIRCLE
2. TRIANGLE
3. TRIGONOMETRIC FUNCTIONS
4. TRIGONOMETRIC IDENTITIES and EQUATIONS
Double Angle Formulae
5. SINE and COSINE FUNCTIONS
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1
CIRCLE
Equation of a circle: centre  xC , yC  and radius a
Circumference
Area
 x  xC 
2
  y  yC   a 2
2
C  2 a
A   a2
In many scientific and engineering calculations radians are used in preference to
degrees in the measurement of angles. An angle of one radian is subtended by an arc
having the same length as the radius.
1 revolution  360o  2 rad
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The length of an arc s of a circle which subtends an angle  is
s  a
The ratio of the area of the sector to the area of the full circle is the
same as the ratio of the angle  to the angle in a full circle. The full
circle has area  a 2 . Therefore
area of sector 

area of circle 2
a2
area of sector  
2
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TRIANGLE
The Theorem of Pythagoras
c 2  a 2  b2
Trigonometrical ratios in a right-angled triangle
Sine ratio
sin( ) 
a
c
Cosine ratio
cos( ) 
b
c
Tangent ratio
tan( ) 
a
b
Cosecant ratio
cosec( ) 
Scant ratio
sec( ) 
Cotangent ratio
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cot( ) 
1
c

sin( ) a
1
c

cos( ) b
1
b

tan( ) a
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 or 
sin
cos
tan
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0
0
1
0
30o  / 6 rad 45o  / 4 rad 60o  / 3 rad 90o  / 2 rad
1
1/ 2
1/ 2
3/2
0
1/ 2
1/ 2
3/2
1

1/ 3
3
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Law of Sines
a
b
c


sin( A) sin( B) sin(C )
sin( A) sin( B) sin(C )


a
b
c
Law of Cosines
a 2  b2  c 2
c  a  b  2ab cos(C ) cos(C ) 
2a b
2
2
2
b2  c 2  a 2
a  b  c  2b c cos( A) cos( A) 
2b c
2
2
2
a 2  c 2  b2
b  a  c  2ac cos( B ) cos( B ) 
2a c
2
2
2
Law of Tangents
ab
tan 

a b
2 


ab
ab
tan 

 2 
Area
A
1
2
 base  height 
A  12 b h
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TRIGONOMETRIC FUNCTIONS
Knowledge of the trigonometric functions is vital in very many fields of engineering,
mathematics and physics.
Sine function
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y  sin  
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Cosine function
y  cos  
EVEN FUNCTION cos( )  cos( )
spacing between blue dots is 30o
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Tangent function
y  tan  
ODD FUNCTION tan( )   tan( )
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TRIGONOMETRIC IDENTITIES and EQUATIONS
sin 2 ( )  cos2 ( )  1
sec2 ( )  1  tan 2 ( )
cosec 2 ( )  1  cot 2 ( )
sin( x )  1  cos2 ( x )
tan 2 ( x)  1 
cos( x )  1  sin 2 ( x )
sin 2 ( x)
sin 2 ( x)  cos2 ( x)
1

1


2
2
cos ( x)
cos ( x)
cos2 ( x)
sin(   )  sin( ) cos( )  cos( )sin( )
sin(   )  sin( ) cos( )  cos( )sin( )
cos(   )  cos( ) cos( )  sin( )sin( )
cos(   )  cos( ) cos( )  sin( )sin( )
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tan(   ) 
tan( )  tan( )
1  tan( ) tan( )
tan(   ) 
tan( )  tan( )
1  tan( ) tan( )
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  
  
sin( )  sin( )  2sin 
 cos 

 2 
 2 
     
cos( )  cos( )  2sin 
 sin 

 2   2 
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Double angle formulae
sin(2 x)  2sin( x)cos( x)
cos(2 x )  cos2 ( x )  sin 2 ( x )  2cos 2 ( x )  1 cos 2 ( x) 
sin(2 x )
2sin( x ) cos( x )

cos(2 x ) cos2 ( x )  sin 2 ( x )
2 tan( x )
tan(2 x ) 
1  tan 2 ( x )
tan(2 x ) 
1
2
1  cos(2 x) 
 cos2 ( x )
 sin 2 ( x )   1 
cos(2 x )  cos2 ( x )  sin 2 ( x )  cos 2 ( x ) 1 
1  tan 2 ( x)
 2

2
 cos ( x )   sec ( x ) 
1  tan 2 ( x )
cos(2 x ) 
1  tan 2 ( x )


2sin( x ) cos2 ( x ) 2 tan x
sin(2 x )  2sin( x ) cos( x ) 

cos( x )
sec2 ( x )
2 tan x
sin(2 x ) 
1  tan 2 ( x )
The substitution t  tan( x / 2) is often a useful one for integration of trigonometric
functions because we can express
2t
sin( x ) 
1 t2
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1 t2
cos( x ) 
1 t2
dx 
2 dt
1 t2
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SINE and COSINE FUNCTIONS
sin(   )  sin( )cos( )  cos( )sin( )
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  90o   / 2 rad sin(   / 2)  cos( )
 sine curve shifted to left through /2 rad
  90o   / 2 rad sin(   / 2)   cos( )
 sine curve shifted to right through /2 rad
  180o   rad sin(   )   sin( )
 sine curve shifted to left through  rad
  180o   rad sin(   )   sin( )
 sine curve shifted to right through  rad
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cos(   )  cos( )cos( )  sin( )sin( )
  90o   / 2 rad cos(   / 2)   sin( )
 cosine curve shifted to left through /2 rad
  90o   / 2 rad cos(   / 2)  sin( )
 cosine curve shifted to right through /2 rad
  180o   rad cos(   )   cos( )
 cosine curve shifted to left through  rad
  180o   rad cos(   )   cos( )
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 cosine curve shifted to right through  rad
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