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Understanding Spherical Coordinates: Triangles and Trigonometry

Dive into spherical coordinates using triangles and trigonometry, exploring points and labels to enhance comprehension.

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Understanding Spherical Coordinates: Triangles and Trigonometry

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  1. Spherical coordinates z y x

  2. Spherical coordinates z y x P(x,y,z)

  3. Spherical coordinates z y x P(x,y,z) P’(x,y,0)

  4. Spherical coordinates z y x P(x,y,z) x y P’(x,y,0)

  5. Spherical coordinates z y x P(x,y,z) x r y P’(x,y,0)

  6. Spherical coordinates z y x P(x,y,z) x r y P’(x,y,0)

  7. Spherical coordinates z y x P(x,y,z) x r y P’(x,y,0)

  8. Spherical coordinates z y x Q P(x,y,z) x r y P’(x,y,0)

  9. Spherical coordinates z y x Q P(x,y,z) ? ? x r y P’(x,y,0)

  10. Spherical coordinates z y x Let’s label the origin O,and consider the triangles OPQ and OPP’ Q P(x,y,z) ? O x r y P’(x,y,0)

  11. Spherical coordinates z y x Right triangle trig gives us everything we need to know! Q P(x,y,z) ? O x r y P’(x,y,0)

  12. Spherical coordinates z y x Q P(x,y,z) ? ? z O x r y P’(x,y,0)

  13. Let’s pull out that triangle for a closer look z

  14. Let’s pull out that triangle for a closer look z

  15. Spherical coordinates z y x Q P(x,y,z) ? ? O x r y P’(x,y,0)

  16. Spherical coordinates z y x Q P(x,y,z) ? ? O x r y P’(x,y,0)

  17. So we know: And since

  18. Spherical Coordinates

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