How will you prove the formula sin(A+B)=sinAcosB+cosAsinB using formula of scalar product of two vectors?

1 Answer
Cesareo R.
Oct 2, 2016

See below.

Explanation:

Given

vec v_(alpha) = (sin alpha, cos alpha) and
vec v_(beta) = (sin beta, cos beta)

both unit vectors, because

norm( vec v_(alpha)) = norm( vec v_(beta)) =1

Have their inner product given by

<< vec v_(alpha), vec v_(beta) >> = sin alpha sin beta+cos alpha cos beta and which is the projection on vec v_(alpha) onto vec v_(beta) or vice-versa.

This is exactly cos(alpha-beta) = cos(beta-alpha)

If we take complex numbers and using the de Moivre's identity which reads

e^(ix) = cos x + i sin x we have

(cos alpha + i sin alpha)(cos beta + i sin beta) = e^(ialpha)e^(ibeta) =cosalpha cosbeta-sinalpha sinbeta+i(sinalpha cosbeta+cosalpha sinbeta) =e^(i(alpha+beta)) = cos(alpha+beta)+isin(alpha+beta).

In the case of vectors we have projections and in the case of complex numbers we have rotations.

Using vectors with additional phase additions we can obtain also

sin alpha cos beta + cosalpha sinbeta=sin(alpha+beta)

It is left as an exercise.

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