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or ", Cell[BoxData[ \(TraditionalForm\`cos(\[Theta]) = 1\/2\)]], ", yielding ", Cell[BoxData[ \(TraditionalForm\`\[Theta] = 0\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\[Theta] = \[Pi]/3\)]], " or \[Theta]=2\[Pi]/3. These are the simultaneous points of intersection \ and have corresponding rectangular coordinates ", Cell[BoxData[ \(TraditionalForm \`\((x, y)\) = \(\((2 \( sin(0)\) \(cos(0)\), 2 \( sin(0)\) \(sin(0)\))\) = \((0, 0)\)\)\)]], ", ", Cell[BoxData[ \(TraditionalForm \`\((x, y)\) = \(\((2 \( sin(\[Pi]/3)\) \(cos(\[Pi]/3)\), 2 \( sin(\[Pi]/3)\) \(sin(\[Pi]/3)\))\) = \((\@3\/2, 3\/2)\)\)\)]], ", and ", Cell[BoxData[ \(TraditionalForm \`\((x, y)\) = \(\((2 \( sin(2 \[Pi]/3)\) \(cos(2 \[Pi]/3)\), 2 \( sin(2 \[Pi]/3)\) \(sin(2 \[Pi]/3)\))\) = \((\(-\(\@3\/2\)\), 3\/2)\)\)\)]], ".\n Note that the second curve passes through the origin for other values \ of \[Theta], ", Cell[BoxData[ \(TraditionalForm\`\[Theta] = \[Pi]/2\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\[Theta] = \[Pi]\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`\[Theta] = 3 \[Pi]/2\)]], "." }], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["41.", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(PolarPlot[{1, 2 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Since the first equation does not \ involve \[Theta], we don't have to worry about whether the intersetions are \ \"simultaneous\" or not. We simply need to solve ", Cell[BoxData[ \(TraditionalForm\`2 \( sin(2 \[Theta])\) = 1\)]], ". This gives ", Cell[BoxData[ \(TraditionalForm \`2 \[Theta] = \(\(sin\^\(-1\)\)(1/2) = \[Pi]\/6 + \ 2\ n\ \[Pi]\)\)]], ", or ", Cell[BoxData[ \(TraditionalForm \`2 \[Theta] = \(\(sin\^\(-1\)\)(1/2) = \(5 \[Pi]\)\/6 + \ 2\ n\ \[Pi]\)\)]], " for any integer ", Cell[BoxData[ \(TraditionalForm\`n\)]], ". Thus, ", Cell[BoxData[ \(TraditionalForm\`\[Theta] = \ \[Pi]\/12 + n\ \[Pi]\)]], ", or ", Cell[BoxData[ \(TraditionalForm\`\[Theta] = \ \(5 \[Pi]\)\/12 + n\ \[Pi]\)]], ". 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