Download Trigonometry and Inverse Trigonometry

Maths Extension 1 – Trigonometry
Trigonometry
 Trigonometric Ratios
 Exact Values & Triangles
 Trigonometric Identities
 ASTC Rule
 Trigonometric Graphs
 Sine & Cosine Rules
 Area of a Triangle
 Trigonometric Equations
 Sums and Differences of angles
 Double Angles
 Triple Angles
 Half Angles
 T – formula
 Subsidiary Angle formula
 General Solutions of Trigonometric Equations
 Radians
 Arcs, Sectors, Segments
 Trigonometric Limits
 Differentiation of Trigonometric Functions
 Integration of Trigonometric Functions
 Integration of sin2x and cos2x
 INVERSE TRIGNOMETRY
 Inverse Sin – Graph, Domain, Range, Properties
 Inverse Cos – Graph, Domain, Range, Properties
 Inverse Tan – Graph, Domain, Range, Properties
 Differentiation of Inverse Trigonometric Functions
 Integration of Inverse Trigonometric Functions
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Maths Extension 1 – Trigonometry
Trigonometric Ratios
θ
hypotenuse
adjacent
opposite
hypotenuse
θ
adjacent
opposite
Sine
sin 
=
Cosine
cos 
=
Tangent
tan 
=
Cosecant
cosec 
=
Secant
sec 
=
Cotangent
cot 
=
sin 
cos 
tan 
cosec 
sec 
cot 
=
=
=
=
=
=
sin 
cos
=
=
=
cos90   
sin 90   
cot 90   
sec90   
cosec90   
tan 90   
60’’ = 1’
60’ = 1°
60 seconds = 1 minute
60 minutes = 1 degree
tan  
1
sin 
1
cos
1
tan 
opposite
hypotenuse
adjacent
hypotenuse
opposite
adjacent
hypotenuse
opposite
hypotenuse
adjacent
adjacent
opposite
cot  
cos
sin 
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Maths Extension 1 – Trigonometry
Exact Values & Triangles
30°
2
1
2
3
45°
60°
1
sin
cos
tan
cos ec
sec
cot
0°
0
1
0
––
1
––
1
30°
60°
45°
1
2
3
2
1
2
1
2
1
2
3
2
1
3
1
3
2
2
3
2
2
3
2
2
1
3
1
sin 2   cos 2 
=1
= 1  sin 2 
= 1  cos 2 
3
90°
1
0
––
1
––
0
180°
0
–1
0
––
–1
––
Trigonometric Identities
cos 
2
sin 2 
= cosec2
2
cot 2  = cosec  – 1
1 = cosec2 – cot 2 
1  cot 2 
tan 2   1
=
2
tan  =
1 =
sec 2 
sec 2   1
sec 2   tan 2 
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Maths Extension 1 – Trigonometry
ASTC Rule
90°
nd
2 Quadrant
1st Quadrant
S
A
T
C
180°
3rd Quadrant
0
360
4th Quadrant
270°
First Quadrant: All positive
sin 
sin 
cos
cos
tan 
tan 
+
+
+
Second Quadrant: Sine positive
sin 180   
sin 
cos180   
– cos
tan 180   
– tan 
+
–
–
Third Quadrant: Tangent positive
sin 180   
– sin 
cos180   
– cos
tan 180   
tan 
–
–
+
Fourth Quadrant: Cosine positive
sin 360   
– sin 
cos360   
cos
tan 360   
– tan 
–
+
–
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Maths Extension 1 – Trigonometry
Trigonometric Graphs
Sine & Cosine Rules
Sine Rule:
a
b
c


sin A sin B sin C
sin A sin B sin C


a
b
c
OR
A
c
b
C
B
a
Cosine Rule:
a 2  b 2  c 2  2bc cos A
A
c
b
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a
Maths Extension 1 – Trigonometry
Area of a Triangle
A  12 ab sin C
 C is the angle
 a & b are the two adjacent sides
C
b
a
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Maths Extension 1 – Trigonometry
Trigonometric Equations
 Check the domain eg. 0    360
 Check degrees ( 0    360 ) or radians ( 0    2 )
 If double angle, go 2 revolutions
 If triple angle, go 3 revolutions, etc…
 If half angles, go half or one revolution (safe side)
Example 1
Solve sin θ =
for 0    360
sin  = 12
 = 30°, 150°
1
2
Example 2
Solve cos 2θ =
for 0    360
cos 2 = 12
2 = 60°, 300°, 420°, 660°
 = 30°, 150°, 210°, 330°
1
2
Example 3
Solve tan 2 = 1 for 0    360
tan 2 = 1

= 45°, 225°
2
 = 90°
Example 4
sin 2  cos  0
2sin  cos  cos
cos 2 sin   1
=0
=0
cos
=  12
 = 210°,
330°
=0
 = 90°,
270°
sin 
Example 5
3sin   cos 2  2
3sin   1  2 sin 2  
= –2
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Maths Extension 1 – Trigonometry
2 sin 2   3 sin   1
2sin   1sin   1
sin 
=0
=0
=  12
 = 210°,
330°
= –1
 = 270°
sin 
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Maths Extension 1 – Trigonometry
Sums and Differences of angles
sin    
= sin  cos   cos sin 
sin    
= sin  cos   cos sin 
cos   
= cos cos   sin  sin 
cos   
= cos cos   sin  sin 
tan    
=
tan    
=
tan   tan 
1  tan  tan 
tan   tan 
1  tan  tan 
Double Angles
= 2sin  cos
sin 2
cos 2
= cos 2   sin 2 
= 1  2 sin 2 
= 2 cos2   1
tan 2
=
sin 2 
= 12 1  cos 2 
= 12 1  cos 2 
cos 2 
2 tan 
1  2 tan 2 
Triple Angles
= 3sin   4 sin 3 
sin 3
= 4 cos3   3 cos
cos 3
tan 3
=
Half Angles
=
sin 
cos
3 tan   tan 3 
1  3 tan 2 
2 sin 2 cos 2
= cos2 2  sin 2 2
= 1  2 sin 2 2
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Maths Extension 1 – Trigonometry
tan 
=
2 cos 2 2  1
=
2 tan 2
1  2 tan 2 2
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Maths Extension 1 – Trigonometry
Deriving the Triple Angles
= sin 2   
sin 3
= sin 2 cos  cos 2 sin 
= 2 sin  cos cos  1  2 sin 2  sin 
= 2 sin  cos 2   sin   2 sin 3 
= 2 sin  1  sin 2    sin   2 sin 3 
= 2 sin   2 sin 3   sin   2 sin 3 
= 3sin   4 sin 3 
cos 3
tan 3
=
=
=
=
=
=
=
cos2   
=
tan 2   
tan 2  tan 
1  tan 2 tan 
2 tan 
 tan 
1 tan 2 
=
=
=
=
_
Normal double angle_
Expand double angle_
Multiply_
Change
sin 2   cos 2   1 _
Simplify_
cos 2 cos  sin 2 sin 
2 cos   1cos  2 sin  cos sin 
2
2 cos3   cos  2 sin 2  cos


2 cos3   cos  2 1  cos 2  cos
2 cos 2   cos  2 cos  2 cos3 
4 cos3   3 cos
 tan 
1  2 1tan
 tan 2 
2 tan   tan   tan 3 
1 tan 2 
1 tan 2   2 tan 2 
1 tan 2 
3
3 tan   tan 
1  3 tan 2 
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Maths Extension 1 – Trigonometry
T – Formulae
Let t = tan 2
sin 
=
sin 
=
cos
=
tan 
=
2 sin 2 cos 2

=
2t
1 t2
1 t2
1 t2
2t
1 t2
Using half angles
_
Divide by “1”

2 sin 2 cos 2
cos 2 2  sin 2 2
sin 2   cos 2   1
=
2 sin 2 cos 2
cos 2 2
cos 2 2  sin 2 2
cos 2 2
Divide top and bottom by
cos 2 
=
=
2 tan 2
1  tan 2 2
2t
1 t2
cos ’
cos
=
cos 2 2  sin 2 2
=
cos 2 2  sin 2 2
cos 2 2  sin 2 2
=
 sin
cos 2 2
cos 2 2  sin 2 2
cos 2 2
=
1  tan 2 2
1  tan 2 2
=
1 t2
1 t2
cos
2 
2
tan 
cancel;
=
sin 
cos
=
2t
1 t 2
1 t 2
1 t 2
=
2t
1 t2
2 
2
sin
cos
becomes tan
http://fatmuscle.cjb.net 12
Maths Extension 1 – Trigonometry
Subsidiary Angle Formula
a sin x  b cos x
a
b
=
=
=
=
R(sin x cos x  cos x sin x)
R sin x cos x  R cos x sin x
 a2
Rcos x
=
=
R 2 cos 2 x
R 2 sin 2 x
 b2
a 2  b2
2
2
=
sin x  cos x  1
R2
Rsin x
tan  
R  a 2  b2
a sin x  b cos x
=C
=C
=C
=C
a sin x  b cos x
a cos x  b sin x
a cos x  b sin x
b
a
R sin( x   )
R sin( x   )
R cos( x   )
R cos( x   )
Example 1
Find x. 3 sin x  cos x  1
R =
2
tan 
3  12
= 4
=2

2 sin( x  30)
sin( x  30)
x  30
x
=
1
3
= 30°
=1
= 12
= 30°, 150°
= 60°, 180°
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Maths Extension 1 – Trigonometry
General Solutions of Trigonometric Equations
Then   n  (1)n
sin   sin 
cos  cos
Then   2n  
tan   tan 
Then   n  
Radians
c
= 180°
1° =
c
180
Arcs, Sectors, Segments
Arc Length
l = r
l
θ
r
θ
r
Area of Sector
A = 12 r 2
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Maths Extension 1 – Trigonometry
Area of Segment
A = 12 r 2   sin  
Segment
θ
r
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Maths Extension 1 – Trigonometry
Trigonometric Limits
sin x
tan x
lim
= lim
x 0
x 0
x
x
=
lim cos x
x0
=1
Differentiation of Trigonometric Functions
d
sin x 
dx
= cos x
d
sin f ( x)
dx
=
f ' ( x) cos f ( x)
d
sin( ax  b) 
dx
=
a cos( ax  b)
d
cos x 
dx
=
 sin x
d
cos f ( x)
dx
=
 f ' ( x) sin f ( x)
d
cos(ax  b)
dx
=
 a sin( ax  b)
d
tan x 
dx
=
sec 2 x
d
tan f ( x)
dx
=
f ' ( x) sec 2 f ( x)
d
tan( ax  b) 
dx
=
a sec 2 (ax  b)
d
sec x
dx
=
sec x. tan x
d
cos ecx
dx
=
 cot x. cos ecx
=
 cos ec 2 x
d
cot x
dx
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Maths Extension 1 – Trigonometry
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Maths Extension 1 – Trigonometry
Integration of Trigonometric Functions
 cos ax dx
=
1
sin ax  c
a
 sin ax dx
=

 sec
=
1
tan ax  c
a
=
x
sin 1    c
a
=
x
 x
cos 1    c __OR__  sin 1    c
a
a
=
1
 x
tan 1    c
a
a
 cos ec ax dx
=
1
 cot ax  c
a
 sec ax. tan ax dx
=
1
sec ax  c
a
 cos ecax.cot ax dx
=
1
 cos ecax  c
a

ax
dx
1
a x
2
dx
2
1

a
2
a x
2
2
2
1
 x2
dx
dx
2
1
cos ax  c
a
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Maths Extension 1 – Trigonometry
Integration of sin2x and cos2x
= 2 cos 2 x  1
cos 2x
= 2 cos 2 x
cos 2x  1
1
cos 2 x  1
= cos 2 x
2
2
= 12  cos 2 x  1 dx
 cos x dx
= 12 12 sin 2 x  x   C
= 14 sin 2 x  12 x  C
 cos
cos 2x
2 sin 2 x
sin 2 x
 sin
2
x
dx
2
x
dx =
1
4
sin 2 x  12 x  C
x
dx =
1
2
x  14 sin 2 x  C
= 1  sin 2 x
= 1 cos 2x
= 12 1  cos 2 x 
= 12  1  cos 2 x  dx
= 12 x  12 sin 2 x   C
= 12 x  14 sin 2 x  C
 sin
2
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Maths Extension 1 – Trigonometry
INVERSE TRIGNOMETRY
Inverse Sin – Graph, Domain, Range, Properties
y
1  x  1

2
x
-2

2

2
y

2
 2
sin 1 ( x)   sin 1 x
Inverse Cos – Graph, Domain, Range, Properties
y
1  x  1

0 y 

2
x
-1
0
1
cos 1 ( x)    cos 1 x
Inverse Tan – Graph, Domain, Range, Properties
y
2

All real x
2
x
 2
-2
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Maths Extension 1 – Trigonometry


2
y

2
tan 1 ( x)   tan 1 x
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Maths Extension 1 – Trigonometry
Differentiation of Inverse Trigonometric Functions
d
sin 1 x 
= 1 2
1 x
dx

d
sin 1 ax
dx


d
sin 1 f ( x)
dx
1
=

a  x2
2
f ' ( x)
=
1  [ f ( x)]2


=



=

=

d
cos 1 x
dx
d
cos 1 ax
dx

d
cos 1 f ( x)
dx

1
1  x2
1
a2  x2
f ' ( x)
1  [ f ( x)]2


=
1
1  x2


=
a
a  x2
=
f ' ( x)
a  [ f ( x)]2
d
tan 1 x
dx
d
tan 1 ax
dx

d
tan 1 f ( x)
dx

2
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Maths Extension 1 – Trigonometry
Integration of Inverse Trigonometric Functions

1
a x
2
1

a
dx
2
a2  x2
2
1
 x2
dx
dx
=
x
sin 1    c
a
=
x
 x
cos 1    c __OR__  sin 1    c
a
a
=
1
 x
tan 1    c
a
a
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