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Ley de los senos<br />
I )<br />
FORMULAS DE TRIGONOMETRIA ESFERICA<br />
sincsinA = sinasinC sinasin B = sin bsinA sin bsinC = sincsin B<br />
Ley de los cosenos para lados<br />
II )<br />
cosa = cosbcosc+ sin bsinccosA cosb = cosccosa + sincsinacosB cosc = cosacosb+ sinasinbcosC Ley de los cosenos para ángulos<br />
III )<br />
IV )<br />
V )<br />
cosA =− cosBcosC+ sin BsinCcosa cosB =− cosCcosA + sinCsinAcosb cosC =− cosAcosB+ sinAsinBcosc sin acosB = cosbsinc− sin bcosccosA sin acosC = coscsinb− sin ccosbcosA sin bcosA = cosasinc− sin acosccosB sin bcosC = coscsina − sin ccosacosB sin ccosA = cosasinb− sin acosbcosC sin ccosB = cosbsina − sin bcosacosC sin Acosb = cosBsinC+ cosCsin Bcosa sin Acosc = cosCsin B+ cosBsinCcosa sin Bcosa = cosAsinC+ cosCsinAcosb sin Bcosc = cosCsinA + cosAsinCcosb sin Ccosa = cosAsin B+ cosBsinAcosc sin Ccosb = cosBsinA +<br />
cosAsin Bcosc
VI )<br />
VII )<br />
sin acotb = cot BsinC+ cosCcosa sin acotc = cot Csin B+ cosBcosa sin bcota = cot AsinC+ cosCcosb sin bcotc = cot CsinA + cosAcosb sin ccota = cot Asin B+ cosBcosc sin ccotb = cot BsinA + cosAcosc sin AcotB = cot bsinc− cosccosA sin AcotC = cot csin b− cosbcosA sin BcotA = cot asinc− cosccosB sin BcotC = cot csina − cosacosB sin CcotA = cot asin b− cosbcosC sin CcotB = cot bsina − cosacosC Ley de las tangentes<br />
VIII )<br />
( A − B)<br />
( a − b)<br />
tan tan<br />
2<br />
=<br />
2<br />
( A + B)<br />
( a + b)<br />
tan tan<br />
2<br />
2<br />
( B−C) ( b−c) tan tan<br />
2 2<br />
=<br />
( B+ C)<br />
( b+ c)<br />
tan tan<br />
2 2<br />
( A − C)<br />
( a − c)<br />
tan tan<br />
2 2<br />
=<br />
( A + C)<br />
( a + c)<br />
tan tan<br />
2 2
Fórmulas de los semiángulos<br />
a + b+ c<br />
s =<br />
2<br />
____________________________________________________________________<br />
IX )<br />
sin<br />
2<br />
A sin( s− b) sin(<br />
s− c)<br />
( ) ( )<br />
2 B sin s− c sin s− a<br />
=<br />
sin =<br />
2 sin bsin c<br />
2 sinasinc sin<br />
2<br />
C sin( s− a) sin(<br />
s− b)<br />
=<br />
2 sin asin b<br />
_____________________________________________________________________<br />
X )<br />
cos<br />
2<br />
A sin() s sin(<br />
s− a)<br />
() ( )<br />
2 B sin s sin s−b =<br />
cos =<br />
2 sin bsinc 2 sin asinc cos<br />
2<br />
C sin() s sin(<br />
s−c) =<br />
2 sin asin b<br />
_____________________________________________________________________<br />
XI )<br />
tan<br />
2<br />
A sin( s− b) sin(<br />
s− c)<br />
=<br />
2 sin() s sin(<br />
s−a) tan<br />
2<br />
tan<br />
C sin( s− a) sin(<br />
s− b)<br />
=<br />
2 sin() s sin(<br />
s−c) 2<br />
B sin( s− c) sin(<br />
s− a)<br />
=<br />
2 sin ssin( s−b) _____________________________________________________________________
Fórmulas de los semilados<br />
A + B+ C<br />
S =<br />
2<br />
_____________________________________________________________________<br />
XII )<br />
sin<br />
2<br />
a cos( S) cos(<br />
S− A)<br />
( ) ( )<br />
2 b cos S cos S−B =<br />
sin =<br />
2 sin BsinC 2 sinAsinC sin<br />
2<br />
c cos( S) cos(<br />
S−C) =<br />
2 sinAsin B<br />
_____________________________________________________________________<br />
XIII )<br />
cos<br />
2<br />
a cos( S− B) cos(<br />
S− C)<br />
( ) ( )<br />
2 b cos S− A cos S− C<br />
=<br />
cos =<br />
2 sin BsinC 2 sinAsinC cos<br />
2<br />
c cos( S− A) cos(<br />
S− B)<br />
=<br />
2 sinasin b<br />
_____________________________________________________________________<br />
XIV )<br />
tan<br />
2<br />
a cos( S) cos(<br />
S−A) =<br />
2 cos( S− B) cos(<br />
S− C)<br />
tan<br />
2<br />
tan<br />
c cos( S) cos(<br />
S−C) =<br />
2 cos( S− A) cos(<br />
S− B)<br />
2<br />
b cos( S) cos(<br />
S−B) =<br />
2 cos( S− A) sin(<br />
S− C)<br />
_____________________________________________________________________
Analogías de Neper<br />
XV )<br />
( A − B)<br />
( a − b)<br />
sin tan<br />
2<br />
=<br />
2<br />
( A + B)<br />
c<br />
sin<br />
tan<br />
2<br />
2<br />
( A − B)<br />
( a + b)<br />
cos tan<br />
2<br />
2<br />
=<br />
( A + B)<br />
c<br />
cos<br />
tan<br />
2<br />
2<br />
( a − b)<br />
( A − B)<br />
sin tan<br />
2<br />
=<br />
2<br />
( a + b)<br />
C<br />
sin<br />
tan<br />
2<br />
2<br />
( a − b)<br />
( A + B)<br />
cos tan<br />
2<br />
2<br />
=<br />
( a + b)<br />
C<br />
cos<br />
tan<br />
2<br />
2<br />
_____________________________________________________________________<br />
XVI )<br />
( A − C)<br />
( a − c)<br />
sin tan<br />
2<br />
2<br />
=<br />
( A + C)<br />
b<br />
sin<br />
tan<br />
2<br />
2<br />
( A − C)<br />
( a + c)<br />
cos tan<br />
2<br />
2<br />
=<br />
( A + C)<br />
b<br />
cos<br />
tan<br />
2<br />
2<br />
( a − c)<br />
( A − C)<br />
sin tan<br />
2<br />
2<br />
=<br />
( a + c)<br />
B<br />
sin<br />
tan<br />
2<br />
2<br />
( a − c)<br />
( A + C)<br />
cos tan<br />
2<br />
2<br />
=<br />
( a + c)<br />
B<br />
cos<br />
tan<br />
2<br />
2<br />
_____________________________________________________________________<br />
XVII )<br />
( B−C) ( b−c) sin tan<br />
2 2<br />
=<br />
( B+ C)<br />
a<br />
sin<br />
tan<br />
2 2<br />
( B−C) ( b+ c)<br />
cos tan<br />
2 2<br />
=<br />
( B+ C)<br />
a<br />
cos<br />
tan<br />
2 2<br />
( b−c) ( B−C) sin tan<br />
2 2<br />
=<br />
( b+ c)<br />
A<br />
sin<br />
tan<br />
2 2<br />
( b−c) cos<br />
2<br />
=<br />
( b+ c)<br />
cos<br />
2<br />
( B+ C)<br />
tan<br />
2<br />
A<br />
tan<br />
2
Fórmulas de Gauss<br />
XVIII )<br />
( a − b)<br />
( A − B)<br />
sin sin<br />
2<br />
=<br />
2<br />
c C<br />
sin cos<br />
2<br />
2<br />
( a − b)<br />
( A + B)<br />
cos sin<br />
2<br />
=<br />
2<br />
c C<br />
cos cos<br />
2<br />
2<br />
( a + b)<br />
( A − B)<br />
sin cos<br />
2<br />
2<br />
=<br />
c C<br />
sin sin<br />
2<br />
2<br />
( a + b)<br />
( A + B)<br />
cos cos<br />
2<br />
2<br />
=<br />
c C<br />
cos sin<br />
2<br />
2<br />
_____________________________________________________________________<br />
XIX )<br />
( a − c)<br />
( A − C)<br />
sin sin<br />
2 2<br />
=<br />
b B<br />
sin cos<br />
2 2<br />
( a − c)<br />
( A + B)<br />
cos sin<br />
2 2<br />
=<br />
b B<br />
cos cos<br />
2 2<br />
( a + c)<br />
( A − C)<br />
sin cos<br />
2<br />
2<br />
=<br />
b B<br />
sin sin<br />
2 2<br />
( a + c)<br />
( A + C)<br />
cos cos<br />
2<br />
2<br />
=<br />
b B<br />
cos sin<br />
2<br />
2<br />
_____________________________________________________________________<br />
XX )<br />
( b−c) ( B−C) sin sin<br />
2 2<br />
=<br />
a A<br />
sin cos<br />
2 2<br />
( b−c) ( B+ C)<br />
cos sin<br />
2 2<br />
=<br />
a A<br />
cos cos<br />
2 2<br />
( b+ c)<br />
( B−C) sin cos<br />
2<br />
2<br />
=<br />
a A<br />
sin sin<br />
2 2<br />
( b+ c)<br />
( B+ C)<br />
cos cos<br />
2<br />
2<br />
=<br />
a A<br />
cos sin<br />
2 2