IF WE ARE GIVEN WITH TWO OR MORE DISPLACEMENT EQUATIONS OFS.H.M OF FORMy=Asin(wt+kx+phase angle)THEN HOW TO FIND RESULTANT AMPLITUDE,PHASE ANGLE

Dear Aman

let

y1=A1sin(wt+kx)

y2=A2sin(wt+kx+Φ)

y=y1 +y2

  =A1sin(wt+kx)  +  A2sin(wt+kx+Φ)

  =A1sin(wt+kx )  +A2sin(wt+kx)cos(Φ) +A2cos(wt+kx)sin(Φ)

  =(A1 +A2 cosΦ  )sin(wt+kx ) +A2cos(wt+kx)sin(Φ)

   let  (A1 +A2 cosΦ  ) =A cosδ

and A2 sinΦ  =Asinδ

so y= A cosδ sin(wt+kx ) +Asinδcos(wt+kx)

        =Asin(wt+kx+δ )

  here A is resultant magnitude and  δ is phase angle

A=(A1² +A2² +2A1A2cosδ)^1/2

tanδ =A2sinΦ/(A1 +A2cosΦ)

Dear Aman

let

y1=A1sin(wt+kx)

y2=A2sin(wt+kx+Φ)

y=y1 +y2

  =A1sin(wt+kx)  +  A2sin(wt+kx+Φ)

  =A1sin(wt+kx )  +A2sin(wt+kx)cos(Φ) +A2cos(wt+kx)sin(Φ)

  =(A1 +A2 cosΦ  )sin(wt+kx ) +A2cos(wt+kx)sin(Φ)

   let  (A1 +A2 cosΦ  ) =A cosδ i don''t understand why??

and A2 sinΦ  =Asinδ i don''t understand why?? please help me! 

so y= A cosδ  sin(wt+kx ) +Asinδcos(wt+kx)

        =Asin(wt+kx+δ )

  here A is resultant magnitude and  δ is phase angle

A=(A1² +A2² +2A1A2cosδ)^1/2

tanδ =A2sinΦ/(A1 +A2cosΦ)