3.8: Trigonometrical Formulas
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I gather here merely for reference a set of commonly-used trigonometric formulas. It is a matter of personal preference whether to commit them to memory. It is probably fair to remark that anyone who is regularly engaged in problems in celestial mechanics or related disciplines will be familiar with most of them, at least from frequent use, whether or not any conscious effort was made to memorize them. At the very least, the reader should be aware of their existence, even if he or she has to look to recall the exact formula.
sinAcosA=tanA
sin2A+cos2A=1
1+cot2A=csc2A
1+tan2A=sec2A
secAcscA=tanA+cotA
sec2Acsc2A=sec2A+csc2A
sin(A±B)=sinAcosB±cosAsinB
cos(A±B)=cosAcosB∓sinAsinB
tan(A±B)=tanA±tanB1∓tanAtanB
sin2A=2sinAcosA
cos2A=cos2A−sin2A=2cos2A−1=1−2sin2A
tan2A=2tanA1−tan2A
sin12A=√1−cosA2
cos12A=√1+cosA2
tan12A=√1−cosA1+cosA=1−cosAsinA=sinAA+cosA=cscA−cotA
sinA+sinB=2sin12Scos12D,
where S=A+BandD=A−B
sinA−sinB=2cos12Ssin12D
cosA+cosB=2cos12Scos12D
cosA−cosB=−2sin12Ssin12D
sinAsinB=12(cosD−cosS)
cosAcosB=12(cosD+cosS)
sinAcosB=12(sinS+sinD)
sinA=T√1+T2=2T1+t2,
where T=tanA and t=tan12A
cosA=1√1+T2=1−t21+t2
tanA=T=2t1−t2
s=sinA,c=cosA
cosA=csinA=scos2A=2c2−1sin2A=2cscos3A=4c3−3csin3A=3s−4s3cos4A=8c4−8c2+1sin4A=4c(s−2s3)cos5A=16c5−20c3+5csin5A=5s−20s3+16s5cos6A=32c6−48c4+18c2−1sin6A=2c(3s−16s3+16s5)cos7A=64c7−112c5+56c3−7csin7A=7s−56s3+112s5−64s7cos8A=128c8−256c6+160c4−32c2+1sin8A=8c(s−10s3+24s5−16s7)
cos2A=12(cos2A+1)cos3A=14(cos3A+3cosA)cos4A=18(cos4A+4cos2A+3)cos5A=116(cos5A+5cos3A+10cosA)cos6A=132(cos6A+6cos4A+15cos2A+10)cos7A=164(cos7A+7cos5A+21cos3A+35cosA)cos8A=1128(cos8A+8cos6A+28cos4A+56cos2A+35)
sin2A=12(1−cos2A)sin3A=14(3sinA−sin3A)sin4A=18(cos4A−4cos2A+3)sin5A=116(sin5A−5sin3A+10sinA)sin6A=132(10−15cos2A+6cos4A−cos6A)sin7A=164(35sinA−21sin3A+7sin5A−sin7A)sin8A=1128(cos8A−8cos6A+28cos4A−56cos2A+35)
sinA=A−A33!+A55!−...
cosA=1−A22!+A44!−...
∫π/20sinmθcosnθdθ=(m−1)!!(n−1)!!X(m+n)!!,where X=π/2 if m and n are both even, and X=1 otherwise.
eniθ=einθ (de Moivre's theorem - the only one you need know. All others can be deduced from it.)
Plane triangles:
asinA=bsinB=csinC
a2=b2+c2−2bccosA
acosB+bcosA=c
s=12(a+b+c)
sin12A=√(s−b)(s−c)s(s−a)
cos12A=√s(s−a)bc
tan12A=√(s−b)(s−c)s(s−a)
Spherical triangles
sinasinA=sinbsinB=sincsinC
cosa=cosbcosc+sinbsinccosA
cosA=−cosBcosC+sinBsinCcosa
cos(IS)cos(IA)=sin(IS)cot(OS)−sin(IA)cot(OA)