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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.

(II) Degrees Radians

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

is set to radians when you are making radian

calculations.

(III) Definition of trigonometric ratios

y

P(x, y)

r

y

?

x

x

Do not write cos-1q, tan-1q .

Graph of ysin x

Graph of ycos x

Graph of ytan x

From the above definitions, the signs of sin ?,

cos ? tan ? in different quadrants can be

obtained. These are represented in the following

diagram

(IV) Trigonometrical ratios of special angles

- What are special angles?

Trigonometrical ratios of these angles are worth

exploring

0

p

2p

sin 360 0

sin 180 0

sin 0 0

sin 90 1

sin 270 -1

0

p

2p

cos 360 1

cos 180 -1

cos 0 1

cos 270 0

cos 90 0

p

0

2p

tan 360 0

tan 180 0

tan 0 0

tan 270 is undefined

tan 90 is undefined

Using the equilateral triangle (of side length 2

units) shown on the right, the following exact

values can be found.

Complete the table. What do you observe?

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Important properties

Important properties

or 2p q

or - q

In the diagram, q is acute. However, these

relationships are true for all sizes of q.

Complementary angles

E.g. 30 60 are complementary angles.

Recall

We say that sine cosine are complementary

functions.

Also, tangent cotangent are complementary

functions.

E.g. 1 Simplify (i) sin 210? (ii) cos

(iii) tan( ) (iv) sin( )

Solution

sin(18030?)

(a) sin 210?

- sin 30?

3rd quadrant

210 18030

4th quadrant

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

(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

? ?

? ? 180 or ? ? ?

? 360 - ? or ? 2? - ?

E.g.

(1st quadrant)

(1st quadrant)

E.g.

(2nd quadrant)

(2nd quadrant)

E.g.

(3rd quadrant)

(3rd quadrant)

E.g.

(4th quadrant)

(4th quadrant)

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

E.g. 3(a) sin . Solve for

? if

E.g. 3(b) cos .

Solve for ? if

Basic angle, a 36.870o

(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

(VI) 3 Important Identities

Dividing (1) throughout by cos2 A,

tan 2 x (tan x)2

Dividing (1) throughout by sin2 A,

(VII) Important Formulae

(1) Compound Angle Formulae

- 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,

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 ?

(2) Double Angle Formulae

(ii) cos 2A cos2 A sin2 A

2 cos2 A 1

1 2 sin2 A

(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

(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

(i) cos 4A (ii) sin ½A

Solution

(i)

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E.g. 6 Prove the following identities

Solution

RHS

E.g. 6 Prove the following identities (ii)

Solution

RHS

E.g. 6 Prove the following identities (iii)

Solution

RHS

E.g. 6 Prove the following identities (iv)

Solution

RHS

(5) The Factor Formulae (Sum or difference of

similar trigo. functions)

Recall compound angles formulae

By letting X A B and Y A B, we

obtain the factor formulae

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

E.g. 8 Show that (ii)

Soln

RHS

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?

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