Some Geometry Some Calculus Some Trigonometry

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SUMMARY

The discussion centers on the mathematical relationship expressed as y≈∆φ×secφ, specifically in the context of geometry and calculus. Participants clarify that the correct formulation is likely ∆y≈sec²(φ)∆φ, which relates changes in angle to changes in length on a curved surface. The conversation emphasizes the application of trigonometric principles to derive these relationships, particularly focusing on the tangent lines to circles and their implications for points A and B on a curved surface.

PREREQUISITES
  • Understanding of trigonometric functions, specifically secant (sec) and its properties.
  • Familiarity with calculus concepts, particularly derivatives and their geometric interpretations.
  • Knowledge of geometry related to curves and tangents.
  • Ability to interpret and manipulate mathematical expressions involving angles and lengths.
NEXT STEPS
  • Research the derivation of the secant function and its applications in calculus.
  • Study the concept of tangent lines in geometry and their relationship to curves.
  • Explore the use of differential calculus in understanding changes in geometric figures.
  • Learn about the implications of angular changes on linear dimensions in curved surfaces.
USEFUL FOR

Mathematicians, physics students, and anyone involved in fields requiring a solid understanding of calculus and geometry, particularly in applications involving curves and trigonometric functions.

Wasif Jalal
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Show that y≈∆φ×secφ in the jpeg attached.

or ∆y = sec φ

A and B are points on curved surface, two lines are extended through origin to a line that is tangent to the circle, these points are A' and B', change in Angle will bring a change in length between A' and B'. I need to know how is this do-able.View attachment 7296[ATTACH=
 

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Are you sure you don't mean:

$$\Delta y\approx\sec^2(\varphi)\Delta \varphi$$?
 

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