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4 Elementary FunctionsTrigonometric Functions

§4.21 Identities

Contents

§4.21(i) Addition Formulas

4.21.1 sinu±cosu=2sin(u±14π)=±2cos(u14π).
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) =±cotucotv-1cotu±cotv.
4.21.6 sinu+sinv =2sin(u+v2)cos(u-v2),
4.21.7 sinu-sinv =2cos(u+v2)sin(u-v2),
4.21.8 cosu+cosv =2cos(u+v2)cos(u-v2),
4.21.9 cosu-cosv =-2sin(u+v2)sin(u-v2).
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(u-v)-cos(u+v),
4.21.16 2cosucosv=cos(u-v)+cos(u+v),
4.21.17 2sinucosv=sin(u-v)+sin(u+v).
4.21.18 sin2u-sin2v =sin(u+v)sin(u-v),
4.21.19 cos2u-cos2v =-sin(u+v)sin(u-v),
4.21.20 cos2u-sin2v =cos(u+v)cos(u-v).

§4.21(iii) Multiples of the Argument

4.21.21 sinz2=±(1-cosz2)1/2,
4.21.22 cosz2=±(1+cosz2)1/2,
4.21.23 tanz2=±(1-cosz1+cosz)1/2=1-coszsinz=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)=2cos2z-1=1-2sin2z=cos2z-sin2z=1-tan2z1+tan2z,
4.21.29 tan(2z)=2tanz1-tan2z=2cotzcot2z-1=2cotz-tanz.
4.21.30 sin(3z) =3sinz-4sin3z,
4.21.31 cos(3z) =-3cosz+4cos3z,
4.21.32 sin(4z) =8cos3zsinz-4coszsinz,
4.21.33 cos(4z) =8cos4z-8cos2z+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)=2n-1k=0n-1sin(z+kπn),
n=1,2,3,.

If t=tan(12z), then

4.21.36 sinz =2t1+t2,
cosz =1-t21+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=cosxcoshy-isinxsinhy,
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.