Từ \(R = r = \sqrt {\frac{L}{C}}\rightarrow\)   \({R^2} = {r^2} = {Z_L}.{Z_C}\rightarrow {Z_C} = \frac{{{R^2}}}{{{Z_L}}}\left( * \right)\)

(Vì  \({Z_L} = \omega L;{Z_C} = \frac{1}{{\omega C}}\rightarrow {Z_L}.{Z_C} = \frac{L}{C}\)  )

\({U_{MB}} = n{U_{AM}} \to {Z_{MB}} = n{Z_{AM}} \to {Z_{MB}} = \sqrt 3 {Z_{AM}} \Leftrightarrow {R^2} + Z_C^2 = 3{r^2} + 3Z_L^2\)

\(\Rightarrow Z_C^2 = 2{R^2} + 3Z_L^2\left( {**} \right) \to {\left( {\frac{{{R^2}}}{{{Z_L}}}} \right)^2} = 2{R^2} + 3Z_L^2\)

\(3Z_L^4 + 2{R^2}Z_L^2 – {R^4} = 0 \to Z_L^2 = \frac{{{R^2}}}{3} \to {Z_L} = \frac{R}{{\sqrt 3 }}\)

 và \({Z_C} = R\sqrt 3 \left( {***} \right)\)

Tổng trở \(Z = \sqrt {{{\left( {R + r} \right)}^2} + {{\left( {{Z_L} – {Z_C}} \right)}^2}} = \frac{{4R}}{{\sqrt 3 }}\)

\(\rightarrow \cos \varphi = \frac{{R + r}}{Z} = \frac{{2R}}{{\frac{{4R}}{{\sqrt 3 }}}} = \frac{{\sqrt 3 }}{2} = 0,866\)

Từ \(R = r = \sqrt {\frac{L}{C}} \to\) \({R^2} = {r^2} = {Z_C}.{Z_C}\)

(Vì  \({Z_L} = \omega L;{Z_C} = \frac{1}{{\omega C}} \to {Z_L}.{Z_C} = \frac{L}{C}\) )

\(U_{AM}^2 = U_R^2 + U_C^2 = {I^2}\left( {{R^2} + Z_C^2} \right)\)

\(U_{MB}^2 = U_r^2 + U_L^2 = {I^2}\left( {{r^2} + Z_L^2} \right) = {I^2}\left( {{R^2} + Z_L^2} \right)\)

Xét tam giác OPQ

\(PQ = {U_L} + {U_C}\)

\(P{Q^2} = {\left( {{U_L} + {U_C}} \right)^2} = {I^2}{\left( {{Z_L} + {Z_C}} \right)^2} = {I^2}\left( {Z_L^2 + Z_C^2 + 2{Z_L}{Z_C}} \right) = {I^2}\left( {Z_L^2 + Z_C^2 + 2{R^2}} \right)\left( 1 \right)\)

\(O{P^2} + O{Q^2} = U_{AM}^2 + U_{MB}^2 = 2U_R^2 + U_L^2 + U_C^2 = {I^2}\left( {2{R^2} + Z_L^2 + Z_C^2} \right)\left( 2 \right)\)

Từ (1) và (2) ta thấy  \(P{Q^2} = O{P^2} + O{Q^2} \to\) tam giác OPQ vuông tại O

Từ  \({U_{MB}} = n{U_{AM}} = \sqrt 3 {U_{AM}}\)

\(\tan \left( {\angle POE} \right) = \frac{{{U_{AM}}}}{{{U_{MB}}}} = \frac{1}{{\sqrt 3 }} \to \angle POE = {30^0}\). Tứ  giác OPEQ là hình chữ nhật

\(\angle OQE = {60^0} \to \angle QOE = {30^0}\)

  Do đó góc lệch pha giữa u và i trong mạch: \(\varphi = {90^0} – {60^0} = {30^0}\)

Vì vậy , \(\cos \varphi = \cos {30^0} = \frac{{\sqrt 3 }}{2} = 0,866\)