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Physics LibreTexts

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)=cosAcosBsinAsinB

tan(A±B)=tanA±tanB1tanAtanB

sin2A=2sinAcosA

cos2A=cos2Asin2A=2cos2A1=12sin2A

tan2A=2tanA1tan2A

sin12A=1cosA2

cos12A=1+cosA2

tan12A=1cosA1+cosA=1cosAsinA=sinAA+cosA=cscAcotA

sinA+sinB=2sin12Scos12D,

where S=A+BandD=AB

sinAsinB=2cos12Ssin12D

cosA+cosB=2cos12Scos12D

cosAcosB=2sin12Ssin12D

sinAsinB=12(cosDcosS)

cosAcosB=12(cosD+cosS)

sinAcosB=12(sinS+sinD)

sinA=T1+T2=2T1+t2,

where T=tanA and t=tan12A

cosA=11+T2=1t21+t2

tanA=T=2t1t2

s=sinA,c=cosA

cosA=csinA=scos2A=2c21sin2A=2cscos3A=4c33csin3A=3s4s3cos4A=8c48c2+1sin4A=4c(s2s3)cos5A=16c520c3+5csin5A=5s20s3+16s5cos6A=32c648c4+18c21sin6A=2c(3s16s3+16s5)cos7A=64c7112c5+56c37csin7A=7s56s3+112s564s7cos8A=128c8256c6+160c432c2+1sin8A=8c(s10s3+24s516s7)

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(1cos2A)sin3A=14(3sinAsin3A)sin4A=18(cos4A4cos2A+3)sin5A=116(sin5A5sin3A+10sinA)sin6A=132(1015cos2A+6cos4Acos6A)sin7A=164(35sinA21sin3A+7sin5Asin7A)sin8A=1128(cos8A8cos6A+28cos4A56cos2A+35)

sinA=AA33!+A55!...

cosA=1A22!+A44!...

π/20sinmθcosnθdθ=(m1)!!(n1)!!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+c22bccosA

acosB+bcosA=c

s=12(a+b+c)

sin12A=(sb)(sc)s(sa)

cos12A=s(sa)bc

tan12A=(sb)(sc)s(sa)

Spherical triangles

sinasinA=sinbsinB=sincsinC

cosa=cosbcosc+sinbsinccosA

cosA=cosBcosC+sinBsinCcosa

cos(IS)cos(IA)=sin(IS)cot(OS)sin(IA)cot(OA)


This page titled 3.8: Trigonometrical Formulas is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Jeremy Tatum via source content that was edited to the style and standards of the LibreTexts platform.

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