Finding equation for trigonometric function based on two points

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SUMMARY

The discussion focuses on deriving the equation of a cosine function based on two given points. It establishes that a general cosine function is represented as A*cos(Bx+C)+D, which requires multiple points to determine uniquely. However, by restricting the function to the form A*cos(x+B), a unique solution can be found using the two points (x,y) and (w,z). The equations A*cos(x+B) = y and A*cos(w+B) = z allow for the calculation of A and B, respectively, using trigonometric identities.

PREREQUISITES
  • Understanding of trigonometric functions, specifically cosine functions.
  • Familiarity with solving systems of equations.
  • Knowledge of trigonometric identities, including secant and arccosine.
  • Basic algebra skills for manipulating equations.
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  • Study the properties of cosine functions and their transformations.
  • Learn how to solve systems of equations involving trigonometric functions.
  • Explore the application of trigonometric identities in solving equations.
  • Investigate the implications of degrees of freedom in function determination.
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Mathematicians, physics students, and anyone involved in modeling periodic phenomena using trigonometric functions.

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Is there any standard way to do this? Specifically, I know the equation I'm looking for is a cosine function, and I know two points. Is there a way to find the equation of the cos function which has been translated and compressed given only these two points?
 
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Unfortunately there's not a unique cosine function given only two points. A general cosine function is something like A*cos(Bx+C)+D, where all of the capital letters are degrees of freedom. You need at least one point for each degree of freedom to have a unique function.

If you want to restrict the function to only compression and translation you could consider a function of the form A*cos(x+B), in which case you could determine a unique one. If the two points are (x,y) and (w,z), then you have to solve the system of equations:

A*cos(x+B) = y ==> A = y*sec (x+B)
and
A*cos(w+B) = z ==> B = arccos(z/A) + w
 

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