Trigonometry 1

1
Trigonometry (1)

• Polar coordinates
• Sin and Cos
• Sine Rule
• Cosine Rule

2
Polar Coordinates
y
90
x
0
180
Coordinates are given as- (r, ?)
270
As well as (x,y) any coordinate can be
expressed as the angle anti-clockwise from the
x-axis and a distance from the origin.
3
Polar Coordinates
y
Polar coordinates are given as- (r, ?)
90
P
r
?
x
0
180
270
(r cos ? , r sin ?)
Cos ? A/H x/r , so x r cos ?
Sin ? O/H y/r , so y r sin ?
4
Sine and Cosine Curves
5
All Silly Tom Cats
y
90
Sin is ve
All are ve
r
rest are ve
?
x
0
180
Tan is ve
Cos is ve
rest are ve
rest are ve
270
All Silly Tom Cats sounds really stupid, but I
remember it from school - so it kind-of sticks.
6
Activity

• Turn to page 68 of your Core 2 book and answer
• exercise A

7
Height h2 b2 - x2
Height h2 a2 - (c-x)2 h2 a2 - c2 2cx - x2
C
b2 - x2 a2 - c2 2cx - x2
x2
x2
b2 a2 - c2 2cx
a
b
h
b2 a2 - c2 2c x b cos A
a2 b2 c2 - 2bc cos A
B
A
c
cos A Adj/Hyp x/b x b cos A
The cosine rule proof - wont be examined
8
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
9
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.
10
The Cosine Rule - example
Finding an angle
a2 b2 c2 - 2bc cos A
C
angles
13.5
7.1
sides
?
B
A
8.8
13.52 7.12 8.82 - 2x7.1x8.8 cos A
2x7.1x8.8 cos A 7.12 8.82 - 13.52
A 116o to nearest degree
11
Activity

• Turn to page 71 of your Core 2 book and answer
• exercise B
• 1a)
• 2a)

12
sin A Opp/Hyp h/b
C
h b sin A
sin B Opp/Hyp h/a
h a sin B
a
b
h
h b sin A a sin B
B
A
c
The sine rule proof - wont be examined
13
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
14
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
15
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
16
Activity

• Turn to page 74 of your Core 2 book and answer
• exercise C
• 1a)
• 3a)