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4 Elementary FunctionsTrigonometric Functions

§4.21 Identities

Contents
  1. §4.21(i) Addition Formulas
  2. §4.21(ii) Squares and Products
  3. §4.21(iii) Multiples of the Argument
  4. §4.21(iv) Real and Imaginary Parts; Moduli

§4.21(i) Addition Formulas

4.21.1 sinu±cosu=2sin(u±14π)=±2cos(u14π).
4.21.1_5 Acosu+Bsinu=A2+B2cos(uph(A+Bi)),
A,B,
4.21.2 sin(u±v) =sinucosv±cosusinv,
4.21.3 cos(u±v) =cosucosvsinusinv,
4.21.4 tan(u±v) =tanu±tanv1tanutanv,
4.21.5 cot(u±v) =±cotucotv1cotu±cotv.
4.21.6 sinu+sinv =2sin(u+v2)cos(uv2),
4.21.7 sinusinv =2cos(u+v2)sin(uv2),
4.21.8 cosu+cosv =2cos(u+v2)cos(uv2),
4.21.9 cosucosv =2sin(u+v2)sin(uv2).
4.21.10 tanu±tanv =sin(u±v)cosucosv,
4.21.11 cotu±cotv =sin(v±u)sinusinv.

§4.21(ii) Squares and Products

4.21.12 sin2z+cos2z=1,
4.21.13 sec2z=1+tan2z,
4.21.14 csc2z=1+cot2z.
4.21.15 2sinusinv=cos(uv)cos(u+v),
4.21.16 2cosucosv=cos(uv)+cos(u+v),
4.21.17 2sinucosv=sin(uv)+sin(u+v).
4.21.18 sin2usin2v =sin(u+v)sin(uv),
4.21.19 cos2ucos2v =sin(u+v)sin(uv),
4.21.20 cos2usin2v =cos(u+v)cos(uv).

§4.21(iii) Multiples of the Argument

4.21.21 sinz2=±(1cosz2)1/2,
4.21.22 cosz2=±(1+cosz2)1/2,
4.21.23 tanz2=±(1cosz1+cosz)1/2=1coszsinz=sinz1+cosz.

In (4.21.21)–(4.21.23) Table 4.16.1 and analytic continuation will assist in resolving sign ambiguities.

4.21.24 sin(z) =sinz,
4.21.25 cos(z) =cosz,
4.21.26 tan(z) =tanz.
4.21.27 sin(2z)=2sinzcosz=2tanz1+tan2z,
4.21.28 cos(2z)=2cos2z1=12sin2z=cos2zsin2z=1tan2z1+tan2z,
4.21.29 tan(2z)=2tanz1tan2z=2cotzcot2z1=2cotztanz.
4.21.30 sin(3z) =3sinz4sin3z,
4.21.31 cos(3z) =3cosz+4cos3z,
4.21.32 sin(4z) =8cos3zsinz4coszsinz,
4.21.33 cos(4z) =8cos4z8cos2z+1.

De Moivre’s Theorem

When n

4.21.34 cos(nz)+isin(nz)=(cosz+isinz)n.

This result is also valid when n is fractional or complex, provided that πzπ.

4.21.35 sin(nz)=2n1k=0n1sin(z+kπn),
n=1,2,3,.

If t=tan(12z), then

4.21.36 sinz =2t1+t2,
cosz =1t21+t2,
dz =21+t2dt.

§4.21(iv) Real and Imaginary Parts; Moduli

With z=x+iy

4.21.37 sinz=sinxcoshy+icosxsinhy,
4.21.38 cosz=cosxcoshyisinxsinhy,
4.21.39 tanz=sin(2x)+isinh(2y)cos(2x)+cosh(2y),
4.21.40 cotz=sin(2x)isinh(2y)cosh(2y)cos(2x).
4.21.41 |sinz|=(sin2x+sinh2y)1/2=(12(cosh(2y)cos(2x)))1/2,
4.21.42 |cosz|=(cos2x+sinh2y)1/2=(12(cosh(2y)+cos(2x)))1/2,
4.21.43 |tanz|=(cosh(2y)cos(2x)cosh(2y)+cos(2x))1/2.