Graphical solution to an equation relating tan(x) to a semi semi circle

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The discussion focuses on determining the number of positive roots for the equation √(a² - x²) = tan(x) using graphical methods. Participants have sketched graphs of tan(x) and the semicircle for different radii, noting that an increase in radius correlates with more roots. Questions arise regarding the domain of √(a² - x²) and its extension along the x-axis, as well as the number of tangent curves within that interval. Clarification on these points is sought to establish a relationship between the radius and the number of roots. The conversation emphasizes the need for a deeper understanding of the graphical intersection of these functions.
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Homework Statement


Using graphical means, determine how many positive roots exist, as a function of a, the the following equation.


Homework Equations


√(a2-x2) = tan(x)


The Attempt at a Solution



I've sketched graphs showing tan(x) and the semi circle overlapping for various radii. Obviously at the radius increases the number of roots increases but I have no idea how to find a relationship between them. Any help would be awesome!
 
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What is the domain of sqrt(a2-x2)? How far does it extend along the x axis?
How many tangent curves are there in that interval?

ehild
 

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