Pythagoras Theorem

a2 + b2 = c2

where where cc is the hypotenuse while is the hypotenuse while aa and and bb are the lengths of the other two are the lengths of the other two sides.sides.

c

b

a

Trigo Ratios of Acute angles

O

P

Q

OP

PQ

OP

OQ

OQ

PQ

hypotenuse

oppositesin

hypotenuse

adjacentcos

adjacent

oppositetan

TOA CAH SOH

Applications – Angle of elevation and Angle of depression

Applications – Angle of elevation and Angle of depression

A

B Ca

c b

Proof:

.sin2

1sin

2

1sin

2

1CabBacAbctriangleaofArea

h

Draw a perpendicular line from A to BC. The length of this line is h, which is the height of the triangle ABC.

1

2Area of triangle a h Using formula ½ x base x height

sin

sin

hC

bh b C

1

sin2

Area of triangle a b C

Similarly, by drawing perpendicular lines from B to AC and C to AB, we can derive other versions of the formula

A

B Ca

c b

1 1 1 sin sin sin

2 2 21 1 12 2 2

bc A ac B ab C

abc abc abc

sin sin sinA B C

a b c

Divide each term by ½ abc

The ratios of the Sine of an angle to its opposite side are equal

Sine Rule

A

B Ca

c bh

xa - x

In ∆ABD, using pythagoras theorem:

D

2 2 2( )c h a x

Similarly, in ∆ADC, using pythagoras theorem: 2 2 2b h x

Using the ratio of cosine in ∆ADC: cosx

Cb

cosx b C Eliminating x and h:

2 2 2 22c h a ax x

2 2 2

2 2 2

2

2

2

2

2

2

2 ( cos )

2 cos

c a ax x

c a b a b C

b

c b a ab C

x

Cosine Rule

A

B Ca

c b

2 2 2

2 2 2

2 2 2

2 cos

2 cos

2 cos

a b c bc A

b a c ac B

c b a ab C

2 2 2

2 2 2

2 2 2

cos2

cos2

cos2

b c aA

bc

a c bB

ac

a b cC

ab

The formula can be rearranged to:

Which one to use depends whether the unknown is a length or an angle

Cosine Rule

A

B Ca

c b

( )( )( )Area of a triangle s s a s b s c

where2

a b cs

(1/2 the perimeter of the triangle)

Heron’s Formula

Why use Heron’s Formula?

A

B C9

4 7 Find Area of Triangle ABC.

2 2 2

2 2 2

2 cos

9 4 7 2(4)(7)cos

56cos 16 49 81

16cos

56

106.6015496

a b c bc A

A

A

A

A

o2

1(4)(7)sin106.6015496

213.416...

13.4

Area

units

o

First: Find one of the angles, then use formula for Area of triangle

Why use Heron’s Formula?

A

B C9

4 7 Find Area of Triangle ABC.

4 7 9

210

s

2

( )( )( )

10(10 4)(10 7)(10 9)

180

6 5

13.4

Area s s a s b s c

units

Using Heron’s Formula:

Advantage: Answer is more accurate and can be worked out faster!

When to use Heron’s Formula?

• When ALL 3 sides of the triangle is given/found and you are asked to find AREA

Important Points

• Bearings are measured from the NORTH

• Bearings are measured in Clockwise Direction

• Bearings are written in 3-digit

(e.g: 030°, 032.1°)