x 1 ( t ) = 4 s i n ( ω t ) x_1(t)=4 sin(\omega t) x 1 ( t ) = 4 s in ( ω t ) x 2 ( t ) = 3 s i n ( ω t + π / 2 ) = 3 c o s ( ω t ) x_2(t)=3 sin(\omega t + \pi / 2) = 3 cos(\omega t) x 2 ( t ) = 3 s in ( ω t + π /2 ) = 3 cos ( ω t ) x ( t ) = x 1 ( t ) + x 2 ( t ) = 4 s i n ( ω t ) + 3 c o s ( ω t ) = A ⋅ s i n ( ω t + ϕ ) x(t)=x_1(t)+x_2(t)=4sin(\omega t)+3cos(\omega t)=A \sdot sin(\omega t +\phi) x ( t ) = x 1 ( t ) + x 2 ( t ) = 4 s in ( ω t ) + 3 cos ( ω t ) = A ⋅ s in ( ω t + ϕ ) where
A = 3 2 + 4 2 = 5 A=\sqrt{3^2+4^2}=5 A = 3 2 + 4 2 = 5 ϕ = a r c s i n ( 3 / A ) = a r c c o s ( 4 / A ) \phi=arcsin(3/A)=arccos(4/A) ϕ = a rcs in ( 3/ A ) = a rccos ( 4/ A ) so, the resultant oscillation:
x ( t ) = 5 s i n ( ω t + a r c s i n ( 3 / 5 ) ) x(t)=5 sin(\omega t + arcsin(3/5)) x ( t ) = 5 s in ( ω t + a rcs in ( 3/5 )) second option:
x 2 ( t ) = 3 s i n ( ω t − π / 2 ) = − 3 c o s ( ω t ) x_2(t)=3sin(ωt-π/2)=-3cos(ωt) x 2 ( t ) = 3 s in ( ω t − π /2 ) = − 3 cos ( ω t ) A = ( − 3 ) 2 + 4 2 = 5 A=\sqrt{(-3)^2+4^2}=5 A = ( − 3 ) 2 + 4 2 = 5 ϕ = a r c s i n ( − 3 / A ) = − a r c s i n ( 3 / A ) \phi=arcsin(-3/A)=-arcsin(3/A) ϕ = a rcs in ( − 3/ A ) = − a rcs in ( 3/ A ) then, the resultant oscillation:
x ( t ) = 5 s i n ( ω t − a r c s i n ( 3 / 5 ) ) x(t)=5 sin(\omega t - arcsin(3/5)) x ( t ) = 5 s in ( ω t − a rcs in ( 3/5 ))
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