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 sin
2
x and cos
2
x
 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
1
Maths Extension 1 – Trigonometry
Trigonometric Ratios
Sine sin
θ
=
hypotenuse
opposite
Cosine cos
θ
=
hypotenuse
adjacent
Tangent tan
θ
=
adjacent
opposite
Cosecant cosec
θ
=
θ
sin
1
=
opposite
hypotenuse
Secant sec

θ
=
θ
cos
1
=
adjacent
hypotenuse
Cotangent cot
θ
=
θ
tan
1
=
opposite
adjacent
sin
θ
=
( )
θ
−°90cos
cos
θ
=
( )
θ
−°90sin
tan

θ
=
( )
θ
−°90cot
cosec
θ
=
( )
θ
−°90sec
sec
θ
=
( )
θ
−°90cosec
cot
θ
=
( )
θ
−°90tan
60 seconds = 1 minute 60’’ = 1’
60 minutes = 1 degree 60’ = 1°
θ
θ
θ
cos
sin

tan =
θ
θ
θ
sin
cos
cot =
2
θ
θ
hypotenuse
hypotenuse
opposite
adjacent
adjacent
opposite
Maths Extension 1 – Trigonometry
Exact Values & Triangles
0° 30° 60° 45° 90° 180°
sin
0
2
1
2
3
2
1
1 0
cos
1

2
3
2
1
2
1
0 –1
tan
0
3
1
3
1 –– 0
cos
ec –– 2
3
2
2
1 ––
sec
1
3
2
2
2
–– –1
cot
––
3
3

1
1 0 ––
Trigonometric Identities
θθ
22
cossin +
= 1
θ
2
cos
=
θ
2
sin1−
θ
2
sin
=
θ
2
cos1−
θ
2
cot1+
= cosec
2
θ
θ
2
cot

= cosec
2
θ
– 1
1 = cosec
2
θ
–
θ
2
cot
1tan
2
+
θ
=
θ
2
sec
θ
2
tan
=
1sec
2
−
θ
1 =
θθ
22

tansec −
3
1
1
2
45°
3
1
2
30°
60°
Maths Extension 1 – Trigonometry
ASTC Rule
First Quadrant: All positive
θ
sin
θ
sin
+
θ
cos
θ
cos
+
θ
tan
θ
tan
+
Second Quadrant: Sine positive

( )
θ
−°180sin
θ
sin
+
( )
θ
−°180cos
–
θ
cos
–
( )
θ
−°180tan
–
θ
tan
–
Third Quadrant: Tangent positive
( )
θ
+°180sin
–
θ
sin
–
( )
θ

+°180cos
–
θ
cos
–
( )
θ
+°180tan
θ
tan
+
Fourth Quadrant: Cosine positive
( )
θ
−°360sin
–
θ
sin
–
( )
θ
−°360cos
θ
cos
+
( )
θ
−°360tan
–
θ

tan
–
4
°
°
360
0
90°
180°
270°
S A
T C
1
st
Quadrant
4
th
Quadrant
2
nd
Quadrant
3
rd
Quadrant
Maths Extension 1 – Trigonometry
Trigonometric Graphs
Sine & Cosine Rules
Sine Rule:
C
c

B
b
A
a
sinsinsin
==
OR
c
C
b
B
a
A sinsinsin
==
Cosine Rule:
Abccba cos2
222
−+=
5
A
B
C
a
b
c
A
a
b
c
Maths Extension 1 – Trigonometry

Area of a Triangle
CabA sin
2
1
=
 C is the angle

a
&
b
are the two adjacent sides
6
C
b
a
Maths Extension 1 – Trigonometry
Trigonometric Equations
 Check the domain eg.
°≤≤° 3600
θ
 Check degrees (
°≤≤° 3600
θ
) or radians (
πθ
20 ≤≤
)
 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 =
2
1
for
°≤≤° 3600
θ
θ
sin
=
2
1
θ
= 30°, 150°
Example 2
Solve cos 2θ =
2
1
for
°≤≤° 3600
θ
θ
2cos
=
2
1
θ
2
= 60°, 300°, 420°, 660°
θ

= 30°, 150°, 210°, 330°
Example 3
Solve tan
2
θ
= 1 for
°≤≤° 3600
θ
tan
2
θ
= 1
2
θ
= 45°, 225°
θ
= 90°
Example 4
0cos2sin =+
θθ
θθθ
coscossin2 +
= 0
( )
1sin2cos +
θθ
= 0
θ
cos
= 0

θ
sin
=
2
1
−
θ
= 90°,
270°
θ
= 210°,
330°
Example 5
22cossin3 −=−
θθ
( )
θθ
2
sin21sin3 −−
= –2
1sin3sin2
2
++
θθ
= 0
7
Maths Extension 1 – Trigonometry
( )( )
1sin1sin2 ++
θθ

= 0
θ
sin
=
2
1
−
θ
sin
= –1
θ
= 210°,
330°
θ
= 270°
8
Maths Extension 1 – Trigonometry
Sums and Differences of angles
( )
βα
+sin
=
βαβα
sincoscossin +
( )
βα
−sin
=
βαβα
sincoscossin −

( )
βα
+cos
=
βαβα
sinsincoscos −
( )
βα
−cos
=
βαβα
sinsincoscos +
( )
βα
+tan
=
βα
βα
tantan1
tantan
−
+
( )
βα
−tan
=
βα
βα
tantan1
tantan

+
−
Double Angles
θ
2sin
=
θθ
cossin2
θ
2cos
=
θθ
22
sincos −
=
θ
2
sin21−
=
1cos2
2
−
θ
θ
2tan
=
θ
θ
2
tan21

tan2
−
θ
2
sin
=
( )
θ
2cos1
2
1
−
θ
2
cos
=
( )
θ
2cos1
2
1
+
Triple Angles
θ
3sin
=
θθ
3
sin4sin3 −
θ

3cos
=
θθ
cos3cos4
3
−
θ
3tan
=
θ
θθ
2
3
tan31
tantan3
−
−
Half Angles
θ
sin
=
22
cossin2
θθ
θ
cos
=
2
2
2

2
sincos
θθ
−
=
2
2
sin21
θ
−
=
1cos2
2
2
−
θ
9
Maths Extension 1 – Trigonometry
θ
tan
=
2
2
2
tan21
tan2
θ
θ
−
10

Maths Extension 1 – Trigonometry
Deriving the Triple Angles
θ
3sin
=
( )
θθ
+2sin
=
θθθθ
sin2coscos2sin +
=
( )
θθθθθ
sinsin21coscossin2
2
−+
=
θθθθ
32
sin2sincossin2 −+
=
( )
θθθθ
32
sin2sinsin1sin2 −+−
=
θθθθ
33
sin2sinsin2sin2 −+−

=
θθ
3
sin4sin3 −
_
Normal double angle_
Expand double angle_
Multiply_
Change
1cossin
22
=+
θθ
_
Simplify_
θ
3cos
=
( )
θθ
+2cos
=
θθθθ
sin2sincos2cos −
=
( )
θθθθθ
sincossin2cos1cos2
2
−−

=
θθθθ
cossin2coscos2
23
−−
=
( )
θθθθ
coscos12coscos2
23
−−−
=
θθθθ
32
cos2cos2coscos2 +−−
=
θθ
cos3cos4
3
−
θ
3tan
=
( )
θθ
+2tan
=
θθ
θθ
tan2tan1

tan2tan
−
+
=
θ
θθ
θ
θ
θ
2
2
tan1
tantan2
tan1
tan2
1
tan
−
−
−
+
=
θ
θθ
θ
θθθ
2
22
2
3

tan1
tan2tan1
tan1
tantantan2
−
−−
−
−+
=
θ
θθ
2
3
tan31
tantan3
−
−
11
Maths Extension 1 – Trigonometry
T – Formulae
Let t = tan
2
θ
θ
sin
=
2
1
2
t

t
+
θ
cos
=
2
2
1
1
t
t
+
−
θ
tan
=
2
1
2
t
t
−
θ
sin
=
22
cossin2
θθ
=
2

2
2
2
22
sincos
cossin2
θθ
θθ
+
=
2
2
2
2
2
2
2
2
22
cos
sincos
cos
cossin2
θ
θθ
θ
θθ
+
=
2

2
2
tan1
tan2
θ
θ
+
=
2
1
2
t
t
+
Using half angles
_
Divide by “1”
1cossin
22
=+
θθ
Divide top and bottom by
θ
2
cos
cos
’ cancel;
cos
sin
becomes tan

θ
cos
=
2
2
2
2
sincos
θθ
−
=
2
2
2
2
2
2
2
2
sincos
sincos
θθ
θθ
+
−
=
2
2
2
2

2
2
2
2
2
2
2
2
cos
sincos
cos
sincos
θ
θθ
θ
θθ
+
−
=
2
2
2
2
tan1
tan1
θ
θ
+
−
=

2
2
1
1
t
t
+
−
θ
tan
=
θ
θ
cos
sin
=
2
2
2
1
1
1
2
t
t
t
t
+
−
+

=
2
1
2
t
t
−
12
Maths Extension 1 – Trigonometry
Subsidiary Angle Formula
xbxa cossin +
=
)sincoscos(sin xxxxR +
=
xxRxxR sincoscossin +
a
=
xR cos
2
a∴
=
xR
22
cos
b
=
xR sin
2
b∴
=

xR
22
sin
1cossin
22
=+ xx
=
2
22
R
ba +
22
baR +=
a
b
=
α
tan
xbxa cossin +
= C
)sin(
α
+xR
xbxa cossin −
= C
)sin(
α
−xR
xbxa sincos +
= C

)cos(
α
+xR
xbxa sincos −
= C
)cos(
α
−xR
Example 1
Find x.
1cossin3 =− xx
R =
2
2
13 +
α
tan
=
3
1
=
4
= 2
α
= 30°
)30sin(2 −x
)30sin( −x
30
−
x

x
= 1
=
2
1
= 30°, 150°
= 60°, 180°
13
Maths Extension 1 – Trigonometry
General Solutions of Trigonometric Equations
αθ
sinsin =
Then
απθ
n
n )1(−+=
αθ
coscos =
Then
απθ
±= n2
αθ
tantan =
Then
απθ
+= n
Radians
c
π
= 180°

1° =
180
c
π
Arcs, Sectors, Segments
Arc Length
l
=
θ
r
Area of Sector
A
=
θ
2
2
1
r
14
l
θ
r
θ
r
Maths Extension 1 – Trigonometry
Area of Segment
A
=
( )
θθ

sin
2
2
1
−r
15
θ
r
Segment
Maths Extension 1 – Trigonometry
Trigonometric Limits
x
x
x
sin
lim
0→
=
x
x
x
tan
lim
0→
=
x
x
coslim
0→
= 1

Differentiation of Trigonometric Functions
( )
x
dx
d
sin
=
xcos
[ ]
)(sin xf
dx
d
=
)(cos)(' xfxf
( )
)sin( bax
dx
d
+
=
)cos( baxa +
( )
x
dx
d
cos
=
xsin−
[ ]
)(cos xf

dx
d
=
)(sin)(' xfxf−
( )
)cos( bax
dx
d
+
=
)sin( baxa +−
( )
x
dx
d
tan
=
x
2
sec
[ ]
)(tan xf
dx
d
=
)(sec)('
2
xfxf
( )
)tan( bax

dx
d
+
=
)(sec
2
baxa +
x
dx
d
sec
=
xx tan.sec
ecx
dx
d
cos
=
ecxx cos.cot−
x
dx
d
cot
=
xec
2
cos−
16
Maths Extension 1 – Trigonometry
17

Maths Extension 1 – Trigonometry
Integration of Trigonometric Functions
∫
axcos
dx =
cax
a
+sin
1
∫
axsin
dx =
cax
a
+− cos
1
∫
ax
2
sec
dx =
cax
a
+tan
1
∫
−
22
1
xa

dx =
c
a
x
+






−1
sin
∫
−
−
22
1
xa
dx =
c
a
x
+







−1
cos
__OR__
c
a
x
+






−
−1
sin
∫
+
22
1
xa
dx =
c
a
x
a
+







−1
tan
1
∫
axec
2
cos
dx =
cax
a
+− cot
1
∫
axax tan.sec
dx =
cax
a
+sec
1
∫
axecax cot.cos
dx =
cecax
a
+− cos
1
18

Maths Extension 1 – Trigonometry
Integration of sin
2
x and cos
2
x
x2cos
12cos +x
( )
12cos
2
1
+x
=
1cos2
2
−x
=
x
2
cos2
=
x
2
cos
∫
x
2
cos
dx =

( )
∫
+12cos
2
1
x
dx
=
( )
Cxx ++2sin
2
1
2
1
=
Cxx ++
2
1
4
1
2sin
∫
x
2
cos
dx
=
Cxx ++
2
1

4
1
2sin
x2cos
x
2
sin2
x
2
sin
=
x
2
sin1−
=
x2cos1
−
=
( )
x2cos1
2
1
−
∫
x
2
sin
dx =
( )
∫

− x2cos1
2
1
dx
=
( )
Cxx +− 2sin
2
1
2
1
=
Cxx +− 2sin
4
1
2
1
∫
x
2
sin
dx
=
Cxx +− 2sin
4
1
2
1
19
Maths Extension 1 – Trigonometry

INVERSE TRIGNOMETRY
Inverse Sin – Graph, Domain, Range, Properties
11
≤≤−
x
22
ππ
≤≤− y
xx
11
sin)(sin
−−
−=−
Inverse Cos – Graph, Domain, Range, Properties
11
≤≤−
x
π
≤≤ y0
xx
11
cos)(cos
−−
−=−
π
Inverse Tan – Graph, Domain, Range, Properties
All real x
22
ππ
≤≤− y

xx
11
tan)(tan
−−
−=−
20
2
π
2
π
−
π
2
π
1-1
x
y
0
2-2
x
y
2
-2
x
y
2
π
2
π
−

Maths Extension 1 – Trigonometry
Differentiation of Inverse Trigonometric Functions
( )
x
dx
d
1
sin
−
=
2
1
1
x−
( )
a
x
dx
d
1
sin
−
=
22
1
xa −
( )
)(sin
1
xf

dx
d
−
=
2
)]([1
)('
xf
xf
−
( )
x
dx
d
1
cos
−
=
2
1
1
x−
−
( )
a
x
dx
d
1
cos

−
=
22
1
xa −
−
( )
)(cos
1
xf
dx
d
−
=
2
)]([1
)('
xf
xf
−
−
( )
x
dx
d
1
tan
−
=
2

1
1
x+
( )
a
x
dx
d
1
tan
−
=
22
xa
a
+
( )
)(tan
1
xf
dx
d
−
=
2
)]([
)('
xfa
xf
+

21
Maths Extension 1 – Trigonometry
Integration of Inverse Trigonometric Functions
∫
−
22
1
xa
dx =
c
a
x
+






−1
sin
∫
−
−
22
1
xa
dx =
c
a

x
+






−1
cos
__OR__
c
a
x
+






−
−1
sin
∫
+
22
1
xa
dx =

c
a
x
a
+






−1
tan
1
22