Trigonometry Revision

1
Trigonometry Revision

• Radians
• Area of Sector and Arcs
• Area of triangle
• Sine and Cosine Rules

• Common Identities
• Knowledge of double angle formulae
• Sin and Cos values you should know
• Trig Equations

2
Radians (1)
y
As well as degrees, angles can be expressed in
RADIANS
?/2
1
?
0
x
RADIANS are the distance traveled around the unit
circle.
?
2?
3?/2
0o 0 radians
180o ? radians
360o 2? radians
90o ?/2 radians
270o 3 ?/2 radians
3
Radians (2)
0o 0 radians
90o ?/2 radians
180o ? radians
270o 3 ?/2 radians
360o 2? radians
Example
Example
4
Why Radians (1)
?/2
Its very handy for working out the lengths of
arcs
arc a
r
?
0
?
2?
3?/2
Example
5
Why Radians (2)
?/2
Its very handy for working out the area of
sectors
r
sector
?
0
?
2?
3?/2
Example
6
Formula Book
7
The Area of a Triangle
for working the area in non-right angled
triangles
Area 1/2 ab sin C
C
angles
a
b
sides
B
A
c
Area 1/2 ac sin B
or
Area 1/2 bc sin A
or
8
Area - example
Area 1/2 ab sin C
C
angles
sides
75o
7 cm
4 cm
B
A
Area 1/2 ab sin C
Area 1/2 x 4 x 7 sin 75
Area 14 x 0.966
Area 13.5 cm2 1 d.p.
Example
9
The Sine Rule
is used for working out angles and sides in
non-right angled triangles
It is . a b c
sin A sin B sin C
C
angles
a
b
sides
B
A
c
10
The Sine Rule - example
Finding a side
a b
c sin A sin B sin C
C
angles
sides
4 cm
a ?
75o
35o
B
A
11
The Sine Rule - example
Finding an angle
a b
c sin A sin B sin C
C
angles
sides
6.5 cm
4 cm
85o
?
B
A
4 x sin 85 6.5 x sin A
sin A 4 x sin 85 0.613
6.5
A 38o or 180-38 162o
12
The Cosine Rule
is used for working out angles and sides in
non-right angled triangles
a2 b2 c2 - 2bc cos A
It is .
C
angles
a
b
sides
B
A
c
By similar proofs-
b2 a2 c2 - 2ac cos B
c2 a2 b2 - 2ab cos C
13
The Cosine Rule - example
Finding a side
a2 b2 c2 - 2bc cos A
C
angles
a ?
6
sides
75o
B
A
8
a2 62 82 - 2x6x8 cos 75
a2 36 64 - 96 x 0.2588
a2 75.153
a 8.67 cm 2 d.p.
14
The Cosine Rule - example
Finding an angle
B
angles
13.5
7.1
sides
?
A
C
8.8
C cos-1(-0.4353 ) 116o to nearest degree
15
Trigometrical Identities
Double Angle Formulae
These are all provided in your information
booklet to use
16
Things you should learn (1)
Isosceles triangle - short sides of 1
cos ? adj/hyp 1 / ?2
45
?2
sin ? opp/hyp 1 / ?2
1
45
sin 45 1/?2 0.707
1
cos 45 1/?2 0.707
Using Pythagoras h2 12 12 2 h ?2
17
Things you should learn (2)
Equilateral triangle - sides of 2
cos 60 adj/hyp 1 / 2
cos 30 adj/hyp ?3 / 2
2
2
sin 60 opp/hyp ?3 / 2
?3
sin 30 opp/hyp 1 / 2
60
60
Using Pythagoras 22 12 x2 x2 4 - 1 3 x
?3
18
Things you should learn (3)
cos 30 ?3 / 2
y sin x
sin 60 ?3 / 2
cos 60 1 / 2
sin 30 1 / 2
sin 0 0 sin 90 1
sin 45 1/?2
cos 45 1/?2
y cos x
sin x cos (90-x)
cos x sin (90-x)
cos 0 1 cos 90 0
19
Things you should know - summary
sin 0 0 sin 30 1/2 0.5 sin 45 1/?2
0.707 sin 60 ?3/2 0.866 sin 90 1
y sin x
cos 90 0 cos 60 1/2 0.5 cos 45 1/?2
0.707 cos 30 ?3/2 0.866 cos 0 1
y cos x
20
Equations

• Solve sin (?/2) 1/2 for 0 ? 2?
• sin (?/2) 1/2
• ?/2 sin-1(0.5) ?/6
• Check graph of sine to find other values

By symmetry about ?/2 ?/2 ? - ?/6 5?/6

• ? 2 x ?/2 or ? 2 x 5?/6

• ? ? or ? 5?/3

21
Graphs Sine
22
Equations

• Solve cos2x ½ for 0 x 360
• cos x v(1/2) or - v(1/2 )
• If cos x v(1/2)
• x 45O and ..
• What if cos x -v(1/2) ?
• X 135o and ..

23
Graphs Cosine
24
Equations

• Solve cos2x 1/2
• cos x v(1/2) or - v(1/2 )
• If cos x v(1/2)
• x 45O and ..
• What if cos x -v(1/2) ?
• X 135o and ..

360o - 45o 315o
360o - 135o 225o
25
Where are the 2 solutions for sin, cos and tan
tan ?
cos ?
sin ?
Quadrant 0o to 90o 90o to
180o 180o to 270o 270o to 360o
ve
ve
ve
0 to ?/2
ve
-ve
-ve
?/2 to ?
-ve
-ve
ve
? to 3?/2
-ve
ve
-ve
3?/2 to 2?
26
Which are positive?
?/2
cos
sin
tan
sin
Sine
All
?
0
Tan
Cos
tan
cos
3?/2
27
Which are positive?
?/2
S
A
ll
illy
?
0
C
ats
T
om
3?/2
28
Which are positive?
?/2
S
A
ll
ausages
?
0
C
rap
T
aste
3?/2