paborito ko nga subjects sang highschool....... along with physics and algebra..... (though i hate proving identities....) :lol: :lol: :lol: :lol:

ang mga nasaulo ko nga formula until now...

Sin A / a = Sin B / b = Sin C / c


C^2 = A^2 + B^2 - cos 2ab (daw sala haw hehehehe)

i-derive ang pythagorean theorem a.

ari ho ah....





consider the figure above....
big square with sides a+b
small square with side c
4 units right equilateral triangle with sides a and b

by Geometry:

Area of Big Square (a+B)^2 = Area of small square (c^2) + Area of 4 triangles (4*1/2ab)

a^2 + 2ab + c^2 = c^2 + 2ab

by subtraction:

a^2 + b^2 = c^2

sagad-sagad na gid ya. :) i-derecho abi sa sine law...

indi ina sagad manung idol raymie.... natabunan lang ina sang mga tuktuk over more than 10 years....

ari ang sine law ho ah... after meeting ibutang ko man ang cosine law kay kabalo ko iritan mo man ko dayun cosine law... :lol: :lol: :lol:



Consider Angles C and B with oposite sides c and b respectively

h = c SinB = b SinC

c SinB = b SinC

re arranging:

Sin B / b = Sin C / c

or

b / SinB = c / Sin C


same is true with Either a and b or b and c




mayung aga.. :thumb: :thumb:

san...ta maria elementary school!

baskug gid ya. idol!

:bow: :bow: :bow:

COSINE LAW Derivation...






consider Fig1
c = b CosA + a CosC
multiply each side by c

c^2 = cb CosA + ac CosC ---> let this be equation 1


Consider fig2
a = c CosB +b CosC
multiply each side by a
a^2 = ac CosB + ab CosC ----> let this be equation 2


Consider fig 3
b = a Cos C + c Cos A
multiply each side by b
b^2 = ab CosC + bc CosA ---> let this be equation 3



add equation 2 and 3
a^2 + b^2 = ac CosB + ab CosC + ab CosC + bc CosA
a^2 + b^2 = ac CosB + 2ab CosC +bc CosA
a^2 + b^2 = 2ab CosC + ac CosB + bc CosA

but from equation 1 == ac CosB + bc CosA = c^2


substituting:


a^2 + b^2 = 2abCosC + c^2

re arranging

c^2 = a^2 + b^2 -2abCosC

:naughty:

Hayuf................... :worthy: :worthy: :worthy:


:lol: :lol: :lol:

migo lester... halin ina kay migo bandejas....hehehehe......

migo baltaks... wala ina sa sta maria .... sa san agustin damu ang naka pulos sina... :naughty: :naughty: :naughty: :naughty:

pero basic sciences lang ina.. indi ina pang college....

:lol: :D :D :D :D

baskug gid ah. pang derive tanan mig ah..

Nagatahun ang aton tutorial section! Mapuslan gid sang mga estudyante pa lang. Baskog. Pwede na kaayo! :thumb: :thumb:

migo balts... amu ina ang rasun kung ngaa wala ako naga saulo formula... pwede mo man i derive....
:lol: :lol: :lol: :lol:


more questions manug raymie ah.... para mabaid ta ang mga nagaka tuktukan nga utak... wahehehehe....

migo baltix,

naganosebleed man ko ba...

salamat gid mga manung sg inyo derivations. kay daw absent gid ko sg gintudlo ni.

anu nga manung?.... indi ka sina absent... ato kamu sa basketball court ni badong naga palo palo.... si daryll naman naga pangita pusta para may 10% sya :lol: :lol: :lol: :lol:


balik sa topic ah.... :thumb:

ay oo gali no? pangita pusta si daryl pero maninguha gihapon para siya referee. doble kita sg tigulang. hehehehe...:OT

sige balik sa topic ah.

hehehehe...... :OT na kita ho..... :smoke


gai ko inug derive bi... lantawun ko kung kaya ko pa..... :lol: :lol: :lol: :lol:

anu ang PDE man?.... :smoke

dasig ba, gaedit pa lang ko duwa na dayon sabat.

topic mig for partial differential eqn ah. kay daw may kinahanglan ko i-review.

ok ah.... ipapilit lang ang pamangkut mo kay lantawun ko kung kaya ko sabtun dayun.... :lol:

Maayo gid ang mga contributions naton tanan para makatuon man kita tanan as well.

ari pa gid...

ngaa 180 degrees ang sum sang internal angles sang triangle?

as far as i can recall..... it came from the history of considering the 360deg in the cartesian plane... it was derived wayback ~250BC by a mathematician with the name that starts with "H".... it was not an assumption but was a DISCOVERY (EURICA!!!) .... means daw wala derivation but an existence....

ambot basi sala man ko... :D :smoke

:rolleyes: tan, cge padayon ah,,,tupad ko karon cmo ha?hehehehe.... :cheers: ...
daw sa nalimas man mga balas kag bato nga nakapatong sa likod ko ba... may asin pa gid....

:woot: :woot: :palakan :palakan those were the days.... :OT labay lang ko ah... basi madibdiban ko d..sori po... :)

dumduman ko gart nga kung mag test or exam ta kay migo banasig kag bandejas sa lamesa ko sang maestro ginapapungko? wahehehehehe

Hindi man ka galing angayan mag maestro mig he hehe he :naughty: :naughty: :naughty:

:blink: amo na gani nga ang pag paningkamot ko sang una nagalukso...and that's where the ninjas was born...the rest---'twas history.....hehehehe... kuskus

indi gid eh.... galing kay ginaseparar lang ko sang atun nga mga pinalangga nga mestro :thumb: :thumb: :thumb:

:thumb: :worthy: ..

Tani wala ka na lang nila To^ gin pa kwa exam! :lol: :lol: :lol:

notorious kuno nga nagapasa answer mo..... ti tupad kami sa lamesa nila eh... kulang na lang mani kag beer.... :lol: :lol: :lol:

Back to the 180 degrees internal angles.

If we make a square, there should be 90 deg per internal angle. If we make a diagonal, the angle formed by the diagonal will be 45 deg. So the sum of the formed triangles will be 180 deg (90 + 45 + 45). Expanding this concept into any quadrilateral, we can still find the argument valid as each twist of the side decreases one angle but increases another by the same amount.

Pwede na ni nga proof? :thumb:

:thumb: :thumb: :thumb:

Ok na Idol Raymie

pwede gid eh..... :thumb: :thumb: :thumb:

So i-expand ta ang concept. Who can derive the equation for the sum of internal angles of any polygon.

this is what i come up with.... still need to check though.....


theorem applied: symmetry.... given: total angle of a circle is 360..

refer to the simple diagram .....


in a polygon, there are always 3 angles..... and we already know that a traingle will always have a total of 180deg....

assuming a polygon aith sides N...

angle #1 = 360/N

Angle#2 = Angle#3 = 180 - (360/N)

but since Angle#2 is symetric to the angle on the other side, hence internal angle = 180 - (360/N)

on a polygon with sides N, there are N sides..... and there are also N internal angles....

sum of all internal angles = N*(180 - (360/N) = 180N - 360

factoring 180...

Sum = 180*(N-2)



checking:

Square : N=4...

Sum = 180*(4-2) = 180*2 = 360.... tama?


Hexagon = 180*(6-2) = 180*4 = 720.. tama? and so on.....

what if the number of sides become infinite?

i guess........... then its a cricle :lol: