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## Trigonometry Review

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Title: Trigonometry Review

1
Trigonometry Review
(I) Introduction
By convention, angles are measured from the
initial line or the x-axis with respect to the
origin.
If OP is rotated counter-clockwise from the
x-axis, the angle so formed is positive.
But if OP is rotated clockwise from the
x-axis, the angle so formed is negative.
2
Angles are measured in degrees or radians.
Given a circle with radius r, the angle subtended
by an arc of length r measures 1 radian.
• Care with calculator! Make sure your calculator
calculations.

3
(III) Definition of trigonometric ratios
y
P(x, y)
r
y
?
x
x
Do not write cos-1q, tan-1q .
4
Graph of ysin x
5
Graph of ycos x
6
Graph of ytan x
7
From the above definitions, the signs of sin ?,
cos ? tan ? in different quadrants can be
obtained. These are represented in the following
diagram
8
(IV) Trigonometrical ratios of special angles
• What are special angles?

Trigonometrical ratios of these angles are worth
exploring
9
0
p
2p
sin 360 0
sin 180 0
sin 0 0
sin 90 1
sin 270 -1
10
0
p
2p
cos 360 1
cos 180 -1
cos 0 1
cos 270 0
cos 90 0
11
p
0
2p
tan 360 0
tan 180 0
tan 0 0
tan 270 is undefined
tan 90 is undefined
12
Using the equilateral triangle (of side length 2
units) shown on the right, the following exact
values can be found.
13
Complete the table. What do you observe?
14
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15
Important properties
16
Important properties
or 2p q
or - q
In the diagram, q is acute. However, these
relationships are true for all sizes of q.
17
Complementary angles
E.g. 30 60 are complementary angles.
Recall
18
We say that sine cosine are complementary
functions.
Also, tangent cotangent are complementary
functions.
19
E.g. 1 Simplify (i) sin 210? (ii) cos
(iii) tan( ) (iv) sin( )
Solution
sin(18030?)
(a) sin 210?
- sin 30?
210 18030
20
21
E.g. 2 If sin x 0.6, cos x 0.8, find
(a) sin (3p - x) (b) cos (4p x).
Soln
sin (3p - x) sin (2p p - x)
cos (4p x)
cos (2p x)
sin ( p - x)
cos x
sin x
0.8
0.6
22
(V) Basic Angle
The basic angle is defined to be the positive,
acute angle between the line OP its projection
on the x-axis. For any general angle, there is a
basic angle associated with it.
Let a denotes the basic angle.
? 180 - q or ? ? - q
? ?

23
? ? 180 or ? ? ?
? 360 - ? or ? 2? - ?
24
E.g.
25
E.g.
26
E.g.
27
E.g.
28
Principal Angle Principal Range
Example sin? 0.5
Restricting y sin? inside the principal range
makes it a one-one function, i.e. so that a
unique ? sin-1y exists
29
E.g. 3(a) sin . Solve for
? if
30
E.g. 3(b) cos .
Solve for ? if
Basic angle, a 36.870o
31
(VI) 3 Important Identities
By Pythagoras Theorem,
Note sin 2 A (sin A)2 cos 2 A (cos A)2
sin2 A cos2 A 1
32
(VI) 3 Important Identities
Dividing (1) throughout by cos2 A,
tan 2 x (tan x)2
Dividing (1) throughout by sin2 A,
33
(VII) Important Formulae
(1) Compound Angle Formulae
34
• E.g. 4 It is given that tan A 3. Find,
without using calculator,
• (i) the exact value of tan ?, given that tan (?
A) 5
• the exact value of tan ? , given that sin (?
A) 2 cos (? A)

Solution
(i) Given tan (? A) 5 and tan A 3,

35
Solution
(ii) Given sin (? A) 2 cos (? A) tan A
3,
sin ? cos A cos ? sin A 2 cos? cos A sin?
sin A
(Divide by cos A on both sides)
sin ? cos ? tan A 2(cos ? sin? tan A)
sin ? 3cos ? 2(cos ? 3sin? )
5sin ? cos ?
36
(2) Double Angle Formulae
(ii) cos 2A cos2 A sin2 A
2 cos2 A 1
1 2 sin2 A
37
(3) Triple Angle Formulae
(i) cos 3A 4 cos3 A 3 cos A
Proof
cos 3A cos (2A A)
cos 2A cos A sin 2A sin A
( 2cos2A -1)cos A (2sin A cos A)sin A
2cos3A - cos A 2cos A sin2A
2cos3A - cos A 2cos A(1 - cos2A)
4cos3A - 3cos A
38
(ii) sin 3A 3 sin A 4 sin3 A
Proof
sin 3A sin (2A A)
sin 2A cos A cos 2A sin A
(2sin A cos A )cos A (1 2sin2A)sin A
2sin A(1 sin2A) sin A 2sin3A
3sin A 4sin3A
39
(i) cos 4A (ii) sin ½A
Solution
40
(i)
41
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42
E.g. 6 Prove the following identities
Solution
RHS
43
E.g. 6 Prove the following identities (ii)
Solution
RHS
44
E.g. 6 Prove the following identities (iii)
Solution
45
RHS
46
E.g. 6 Prove the following identities (iv)
Solution
RHS
47
(5) The Factor Formulae (Sum or difference of
similar trigo. functions)
Recall compound angles formulae
48
By letting X A B and Y A B, we
obtain the factor formulae
49
E.g. 8 Show that
(i)
Solution
(i) LHS cos ? cos 3? cos 5?
(cos 5? cos? ) cos 3?
2cos 3? cos 2? cos 3?
cos 3? 2cos2? 1
cos 3? 2(2 cos2? 1) 1
cos 3? (4 cos2? 1) RHS
50
E.g. 8 Show that (ii)
Soln
RHS
51
E.g. 8 Show that
(iii) sin ? sin 3? sin 5? sin 7? 16 sin?
cos2? cos2 2?
Soln
(iii) LHS sin ? sin 3? sin 5? sin 7?
(sin 3? sin ? ) (sin 7? sin 5? )
2sin 2? cos? 2sin 6? cos?
2cos? sin 6? sin 2?
52
4 cos? cos 2? sin 4?
4 cos? cos 2? 2 sin 2? cos 2?
8 cos? cos2 2? sin 2?
8 cos? cos2 2? ( 2 sin? cos? )
16 sin? cos2? cos2 2?
RHS