pythagoras theorem a 2 + b 2 = c 2 where c is the hypotenuse while a and b are the lengths of the...
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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°)