Geo5.4.3

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Determine the lengths of the sides of the triangle whose vertices are (-3,{\frac  {\pi }{6}}),(5,{\frac  {5\pi }{6}}),(7,{\frac  {7\pi }{6}})\,

Let the three vertices be A,B,C

Now

AB={\sqrt  {(-3)^{2}+(5)^{2}-2(-3)(5)\cos {\frac  {\pi }{6}}-{\frac  {5\pi }{6}}}}\,

AB={\sqrt  {34+30\cos 30}}={\sqrt  {34+15{\sqrt  {3}}}}\,

BC={\sqrt  {25+49-70\cos {\frac  {5\pi }{6}}-{\frac  {7\pi }{6}}}}\,

BC={\sqrt  {74-70\cos 60}}={\sqrt  {74-35}}={\sqrt  {39}}\,

CA={\sqrt  {49+9+42\cos {\frac  {7\pi }{6}}-{\frac  {\pi }{6}}}}={\sqrt  {58+42\cos \pi }}\,

CA={\sqrt  {58-42}}=4\,


Main Page:Geometry:Polar Coordinates