You are not logged in.
The Cosine Rule with its squares of sides a , b and c cries out for a fusion with the semiperimeter.
First lets turn the semiperimeter into a few substitutions for later use
Where semiperimeter = s = ( a + b + c ) / 2
( a + b + c ) = 2 s
( a + b - c ) =2 ( s – c )
( a - b + c ) =2 ( s – b )
( b + c - a ) =2 ( s – a )
Then our Cosine rule
a² = b² + c² – 2bcCos A
Cos A = ( b² + c² - a² ) / 2bc
1 – Cos A = ( 2bc - b² - c² + a² ) / 2bc
= ( a – b + c )( a + b - c ) / 2bc
= 2 ( s - b )( s – c ) / bc
1 + Cos A = ( 2bc + b² + c² - a² ) / 2bc
= ( a + b + c )( b + c - a ) / 2bc
= 2 s ( s – a ) / bc
Sin² A = ( 1 – Cos a )( 1 + Cos A )
= 4 s ( s – a )( s – b )( s - c ) / b²c²
_______________________
Sin A = 2 ( √ s ( s – a )( s – b )( s - c ) ) / bc
Area of Triangle = ½ . bc Sin A
______________________
= √ s ( s – a )( s – b )( s - c ) Heron's Area Formula
A much more elegant way of deriving Heron's Formula and there is still the Half angle Formula to go :-
______________
Cos ( A / 2 ) = √ ( 1 + Cos A ) / 2
______________
= √ s ( s – a ) / bc
______________
Sin ( A / 2 ) = √ ( 1 - Cos A ) / 2
___________________
= √ ( s – b )( s – a ) / bc
& Sin A = 2 Sin ( A / 2 ) Cos ( A / 2 ) gives the same Sin A result as above
Tan ( A / 2 ) = Sin ( A / 2 ) / Cos ( A / 2 )
_______________________
= √ ( s – b )( s – c ) / s ( s – a )
corrected mistake in half angle formulae
Last edited by wim (2021-04-12 08:01:23)
Phe φ foe fum a Giant Irrational for every 1 . Eye , jay and kay , know I'm here to stay.The
Beanstalk is Fibonacci we must all surely C. What you say fool ? Eh Wot just Maths is Beautifool.
It blows your mind every correspondence U find.
Offline