Answer to Question #151515 in Differential Geometry | Topology for Dolly

Question #151515
Change the following from cylindrical coordinates to rectangular coordinates
1) (5, π/6, 3)
2) (6, π/3, -5)
1
Expert's answer
2020-12-17T18:51:39-0500

Solution: We know that "The cylindrical coordinates are denoted by "(r,\\theta,z)" and rectangular coordinates are denoted by "(x,y,z)" "

To convert from cylindrical coordinates to rectangular coordinates we use the equations

"x=r~cos~\\theta"

"y=r~sin~\\theta" and

"z=z"


(1) Given cylindrical coordinates are "(5, \\frac{\\pi}{6},3)"

"\\therefore (r,\\theta,z)=(5, \\frac{\\pi}{6},3)"

Now, to find rectangular coordinates, we have

"x=r~cos~\\theta=5~cos~(\\frac{\\pi}{6})=5(\\frac{\\sqrt{3}}{2})=\\frac{5\\sqrt{3}}{2}"


"y=r~sin~\\theta=5~sin~(\\frac{\\pi}{6})=5(\\frac{1}{2})=\\frac{5}{2}"

"z=z=3"

Therefore rectangular coordinates "(x,y,z)=(\\frac{5\\sqrt{3}}{2},\\frac{5}{2},3)"

(2) Given cylindrical coordinates are "(6, \\frac{\\pi}{3},-5)"

"\\therefore (r,\\theta,z)=(6, \\frac{\\pi}{3},-5)"

Now, to find rectangular coordinates, we have

"x=r~cos~\\theta=6~cos~(\\frac{\\pi}{3})=6(\\frac{1}{2})=\\frac{6}{2}=3"


"y=r~sin~\\theta=6~sin~(\\frac{\\pi}{3})=6(\\frac{\\sqrt{3}}{2})=\\frac{6\\sqrt{3}}{2}=3\\sqrt{3}"

"z=z=-5"

Therefore rectangular coordinates "(x,y,z)=(3,3\\sqrt{3},-5)"


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