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Question Number 137864 by n0y0n last updated on 07/Apr/21

 if ,      acosα+bcosβ=Acos∅   proof that       asinα+bsinβ=Asin∅

$$\:\mathrm{if}\:, \\ $$$$\:\:\:\:\mathrm{acos}\alpha+\mathrm{bcos}\beta=\mathrm{Acos}\emptyset \\ $$$$\:\mathrm{proof}\:\mathrm{that}\: \\ $$$$\:\:\:\:\mathrm{asin}\alpha+\mathrm{bsin}\beta=\mathrm{Asin}\emptyset \\ $$

Commented by mr W last updated on 07/Apr/21

one can′t prove something which is  wrong.

$${one}\:{can}'{t}\:{prove}\:{something}\:{which}\:{is} \\ $$$${wrong}. \\ $$

Commented by n0y0n last updated on 07/Apr/21

Commented by mr W last updated on 07/Apr/21

this is not the same as your question!  your question is:  if  acosα+bcosβ=Acos∅  then  asinα+bsinβ=Asin∅  for any a,b,A,α,β,φ.

$${this}\:{is}\:{not}\:{the}\:{same}\:{as}\:{your}\:{question}! \\ $$$${your}\:{question}\:{is}: \\ $$$${if}\:\:\mathrm{acos}\alpha+\mathrm{bcos}\beta=\mathrm{Acos}\emptyset \\ $$$${then}\:\:\mathrm{asin}\alpha+\mathrm{bsin}\beta=\mathrm{Asin}\emptyset \\ $$$${for}\:{any}\:{a},{b},{A},\alpha,\beta,\phi. \\ $$

Commented by mr W last updated on 07/Apr/21

you should mean that constant  acosα+bcosβ can be expressed as A cos φ  and constant  asin α+bsin β can be expressed as A sin φ.  this is true.  generally any two constants, say p   and q, can be expressed as  p=A cos φ  q=A sin φ  with A=(√(p^2 +q^2 ))  φ=tan^(−1) (q/p)

$${you}\:{should}\:{mean}\:{that}\:{constant} \\ $$$$\mathrm{acos}\alpha+\mathrm{bcos}\beta\:{can}\:{be}\:{expressed}\:{as}\:{A}\:\mathrm{cos}\:\phi \\ $$$${and}\:{constant} \\ $$$$\mathrm{asin}\:\alpha+\mathrm{bsin}\:\beta\:{can}\:{be}\:{expressed}\:{as}\:{A}\:\mathrm{sin}\:\phi. \\ $$$${this}\:{is}\:{true}. \\ $$$${generally}\:{any}\:{two}\:{constants},\:{say}\:{p}\: \\ $$$${and}\:{q},\:{can}\:{be}\:{expressed}\:{as} \\ $$$${p}={A}\:\mathrm{cos}\:\phi \\ $$$${q}={A}\:\mathrm{sin}\:\phi \\ $$$${with}\:{A}=\sqrt{{p}^{\mathrm{2}} +{q}^{\mathrm{2}} } \\ $$$$\phi=\mathrm{tan}^{−\mathrm{1}} \frac{{q}}{{p}} \\ $$

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