How do I find the 2 slopes at which this parametric function crosses itself? x ( t ) = 2 sin ⁡ ( 2 t ) y ( t ) = 8 sin ⁡ ( t )

Question

How do I find the 2 slopes at which this parametric function crosses itself?  x  (  t  )  =  2  sin  ⁡  (  2  t  )    y  (  t  )  =  8  sin  ⁡  (  t  )

Elianna Douglas · Accepted Answer

Step 1There’s no need to abandon parametric form:            d      y              d      x        =            d      y              /            d      t              d      x              /            d      t        =            8      cos      ⁡      t              4      cos      ⁡      2      t        =            2      cos      ⁡      t              cos      ⁡      2      t          .Now just find (in terms of t) where it crosses itself.

migongoniwt · Accepted Answer

Step 1Let       x    i    =  2  sin  ⁡  2      t    i   and       y    i    =  8  sin  ⁡      t    i    , ,to find the points where the curve crosses it self, we need       x    i    =      x    j   and       y    i    =      y    j   for some   i  ≠  jSo,   sin  ⁡      t    i    =  sin  ⁡      t    j    −  −  −  &amp;gt;  (  1  ) and   sin  ⁡  2      t    i    =  sin  ⁡  2      t    j    −  −  −  &amp;gt;  (  2  )Step 21) gives       t    i    =  n  π  +  (  −  1      )    n        t    j  2) gives,   2      t    i    =  m  π  +  (  −  1      )    m    2      t    j   where m,n are integers.If   n  =  2  q  ,      t    i    =  2  q  π  +      t    j    −  −  &amp;gt;  (  3  )If   n  =  2  q  +  1  ,      t    i    +      t    j    =  (  2  q  +  1  )  π  −  −  −  &amp;gt;  (  4  )If   m  =  2  r  ,  2      t    i    =  2  r  π  +  2      t    j      ⟹        t    i    =  r  π  +      t    j    −  −  −  &amp;gt;  (  5  )If   m  =  2  r  +  1  ,  2      t    i    =  (  2  r  +  1  )  π  −  2      t    j      ⟹        t    i    +      t    j    =  (  2  r  +  1  )      π    2    −  −  −  &amp;gt;  (  6  )We can only combine, (3) and (5), so that       t    i    =  2  s  π  +      t    j

How do I find the 2 slopes at which this parametric function crosses itself? x ( t )

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