Approximation using pythagorean theoremIm looking at the following diagram,Here, I am interested in a relation between a,s,c, assuming that r &gt;&gt; 1 my first thought is to relate then using the pythagorean theorem, c 2 = a 2 + s 2 Is making sure an approximation a good approach and is there an alternate way to relate my variables? Thank you in advance!

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Approximation using pythagorean theoremIm looking at the following diagram,Here, I am interested in a relation between a,s,c, assuming that   r  &amp;gt;&amp;gt;  1 my first thought is to relate then using the pythagorean theorem,      c    2    =      a    2    +      s    2  Is making sure an approximation a good approach and is there an alternate way to relate my variables? Thank you in advance!

Sawyer Day · Accepted Answer

You can apply the cosine rule:      c    2    =  (  a  +  r      )    2    +      r    2    −  2  r  (  a  +  r  )  cos  ⁡  θ  ,,that is:      c    2    =      a    2    +  2  r  (  a  +  r  )  (  1  −  cos  ⁡  θ  )  ..As   θ  =  s      /    r, if   r  ≫  s we can approximate   cos  ⁡  θ  ≈  1  −            1      2            θ    2    =  1  −            1      2            s    2        /        r    2  , which gives:      c    2    =      a    2    +      s    2        (    1    +                  a        r              )    ..If   r  ≫  a the term   a      /    r is negligible and you obtain       c    2    =      a    2    +      s    2  .

Approximation using pythagorean theorem Im looking at the following diagram,

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