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Get same answers using the Solve[] and the ArcCos[] commands?

Mitchell Sandlin

Posted 9 years ago

Hi In using the Solve[] and the ArcCos[], I am getting sort-of two different answers (see attached) and cannot understand why. Would someone please explain to me why, in solving for Phi, I am getting different answers? The desired answer is Pi/4 Solve[Cos[?] == 4/(4 Sqrt[2]), ?] ArcCos[4/(4 Sqrt[2])] `{{? -> ConditionalExpression[-(?/4) + 2 ? C[1], C[1] ? Integers]},` `{? -> ConditionalExpression[?/4 + 2 ? C[1], C[1] ? Integers]}}` `?/4` Thanks Attachments:

POSTED BY: Mitchell Sandlin

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Mariusz Iwaniuk

Mariusz Iwaniuk, engineer, math hobbyist.

Posted 9 years ago

Because is a general solution. Solve[Cos[\[Phi]] == 1/Sqrt[2], \[Phi]] ({{\[Phi]-> ConditionalExpression[-(\[Pi]/4) + 2 \[Pi] C[1], C[1] \[Element] Integers]}, {\[Phi] -> ConditionalExpression[\[Pi]/4 + 2 \[Pi] C[1], C[1] \[Element] Integers]}}) `C[1]` [Element] Integers from ...-5,-4,-3,-2,-1,0,1,2,3,4,5... Cos[#] & /@ Table[-(\[Pi]/4) + 2 \[Pi] n, {n, -5, 5, 1}] ({1/Sqrt[2], 1/Sqrt[2], 1/Sqrt[2], 1/Sqrt[2], 1/Sqrt[2], 1/Sqrt[2],1/Sqrt[2], 1/Sqrt[2], 1/Sqrt[2], 1/Sqrt[2], 1/Sqrt[2]}) Cos[#] & /@ Table[\[Pi]/4 + 2 \[Pi] n, {n, -5, 5, 1}] ({1/Sqrt[2], 1/Sqrt[2], 1/Sqrt[2], 1/Sqrt[2], 1/Sqrt[2], 1/Sqrt[2],1/Sqrt[2], 1/Sqrt[2], 1/Sqrt[2], 1/Sqrt[2], 1/Sqrt[2]}) For interval (0,Pi) Solve[{Cos[\[Phi]] == 1/ Sqrt[2] && 0 < \[Phi] < \[Pi]}, \[Phi]] ({{\[Phi] -> \[Pi]/4}}) For interval (-Pi,0) Solve[{Cos[\[Phi]] == 1/ Sqrt[2] && -Pi < \[Phi] < 0}, \[Phi]] ({{\[Phi] ->- \[Pi]/4}}) . Plot[{Cos[phi], 1/Sqrt[2]}, {phi, -2 Pi, 2 Pi}, Epilog -> {PointSize[Medium], Point[{{-(( 7 \[Pi])/4), 1/Sqrt[2]}, {-(\[Pi]/4), 1/ Sqrt[2]}, {\[Pi]/4, 1/Sqrt[2]}, {(7 \[Pi])/4, 1/Sqrt[2]}}]}, Prolog -> {Line[{{- 2 \[Pi], -0.6}, {- 2 \[Pi], 1}}], Line[{{-(( 7 \[Pi])/4), -0.6}, {-(( 7 \[Pi])/4), 1}}], Line[{{-( \[Pi]/4), -0.6}, {-( \[Pi]/4), 1}}], Line[{{ \[Pi]/4, -0.6}, { \[Pi]/4, 1}}], Line[{{(7 \[Pi])/4, -0.6}, {(7 \[Pi])/4, 1}}], Line[{{ 2 \[Pi], -0.6}, { 2 \[Pi], 1}}], Text[-(( 7 \[Pi])/4), {-(( 7 \[Pi])/4), -0.9}], Text[- 2 \[Pi], {- 2 \[Pi], -0.9}], Text[-( \[Pi]/4), {-( \[Pi]/4), -0.9}], Text[ \[Pi]/4, { \[Pi]/4, -0.9}], Text[( 7 \[Pi])/4, {( 7 \[Pi])/4, -0.9}], Text[ 2 \[Pi], { 2 \[Pi], -0.9}]}, PlotLabels -> "Expressions", AxesLabel -> {phi}]

Because is a general solution.

Solve[Cos[\[Phi]] == 1/Sqrt[2], \[Phi]]
(*{{\[Phi]-> ConditionalExpression[-(\[Pi]/4) + 2 \[Pi] C[1], C[1] \[Element] Integers]}, {\[Phi] -> ConditionalExpression[\[Pi]/4 + 2 \[Pi] C[1], C[1] \[Element] Integers]}}*)

C[1] [Element] Integers from ...-5,-4,-3,-2,-1,0,1,2,3,4,5...

Cos[#] & /@ Table[-(\[Pi]/4) + 2 \[Pi] n, {n, -5, 5, 1}]
 (*{1/Sqrt[2], 1/Sqrt[2], 1/Sqrt[2], 1/Sqrt[2], 1/Sqrt[2], 1/Sqrt[2],1/Sqrt[2], 1/Sqrt[2], 1/Sqrt[2], 1/Sqrt[2], 1/Sqrt[2]}*)
Cos[#] & /@ Table[\[Pi]/4 + 2 \[Pi] n, {n, -5, 5, 1}]
(*{1/Sqrt[2], 1/Sqrt[2], 1/Sqrt[2], 1/Sqrt[2], 1/Sqrt[2], 1/Sqrt[2],1/Sqrt[2], 1/Sqrt[2], 1/Sqrt[2], 1/Sqrt[2], 1/Sqrt[2]}*)

For interval (0,Pi)

 Solve[{Cos[\[Phi]] == 1/ Sqrt[2] && 0 < \[Phi] < \[Pi]}, \[Phi]]
 (*{{\[Phi] -> \[Pi]/4}}*)

For interval (-Pi,0)

 Solve[{Cos[\[Phi]] == 1/ Sqrt[2] && -Pi < \[Phi] < 0}, \[Phi]]
  (*{{\[Phi] ->- \[Pi]/4}}*)

 Plot[{Cos[phi], 1/Sqrt[2]}, {phi, -2 Pi, 2 Pi}, 
  Epilog -> {PointSize[Medium], 
    Point[{{-(( 7 \[Pi])/4), 1/Sqrt[2]}, {-(\[Pi]/4), 1/
       Sqrt[2]}, {\[Pi]/4, 1/Sqrt[2]}, {(7 \[Pi])/4, 1/Sqrt[2]}}]}, 
  Prolog -> {Line[{{- 2 \[Pi], -0.6}, {- 2 \[Pi], 1}}], 
    Line[{{-(( 7 \[Pi])/4), -0.6}, {-(( 7 \[Pi])/4), 1}}], 
    Line[{{-( \[Pi]/4), -0.6}, {-( \[Pi]/4), 1}}], 
    Line[{{ \[Pi]/4, -0.6}, { \[Pi]/4, 1}}], 
    Line[{{(7 \[Pi])/4, -0.6}, {(7 \[Pi])/4, 1}}], 
    Line[{{ 2 \[Pi], -0.6}, { 2 \[Pi], 1}}], 
    Text[-(( 7 \[Pi])/4), {-(( 7 \[Pi])/4), -0.9}], 
    Text[- 2 \[Pi], {- 2 \[Pi], -0.9}], 
    Text[-( \[Pi]/4), {-( \[Pi]/4), -0.9}], 
    Text[ \[Pi]/4, { \[Pi]/4, -0.9}], 
    Text[( 7 \[Pi])/4, {( 7 \[Pi])/4, -0.9}], 
    Text[ 2 \[Pi], { 2 \[Pi], -0.9}]}, PlotLabels -> "Expressions", 
  AxesLabel -> {phi}]

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