# Math Is Fun Forum

Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫  π  -¹ ² ³ °

You are not logged in.

## #1 2021-03-23 02:27:04

wim
Member
Registered: 2021-03-13
Posts: 7

### Elegant Cosine, Heron's and Half Angle Formulae using semiperimeter

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

## Board footer

Powered by FluxBB