Trigonometric equation inversionI am trying to invert the following equation to have it with θ as the subject: y = cos ⁡ θ sin ⁡ θ − cos ⁡ θ − sin ⁡ θI tried both standard trig as well as trying to reformulate it as a differential equation (albeit I might have chosen an awkward substitution). Nothing seems to stick and I keep on ending up with pretty nasty expressions. Any ideas?

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Trigonometric equation inversionI am trying to invert the following equation to have it with   θ as the subject:  y  =  cos  ⁡  θ  sin  ⁡  θ  −  cos  ⁡  θ  −  sin  ⁡  θI tried both standard trig as well as trying to reformulate it as a differential equation (albeit I might have chosen an awkward substitution). Nothing seems to stick and I keep on ending up with pretty nasty expressions. Any ideas?

lesinetzgl5 · Accepted Answer

y  =  cos  ⁡  θ  sin  ⁡  θ  −  cos  ⁡  θ  −  sin  ⁡  θ  y  −  cos  ⁡  θ  sin  ⁡  θ  =  −  cos  ⁡  θ  −  sin  ⁡  θ      y    2    −  2  y  cos  ⁡  θ  sin  ⁡  θ  +      cos    2    ⁡  θ      sin    2    ⁡  θ  =  1  +  2  cos  ⁡  θ  sin  ⁡  θ      y    2    −  y  sin  ⁡  (  2  θ  )  +      1    4        sin    2    ⁡  (  2  θ  )  =  1  +  sin  ⁡  (  2  θ  )This is a quadratic equation for   sin  ⁡  (  2  θ  ) that you can solve.

klesstilne1 · Accepted Answer

Rewrite the equation as   (  1  −  cos  ⁡  θ  )  (  1  −  sin  ⁡  θ  )  =  1  +  y  .Now make the Weierstrass substitution   t  =  tan  ⁡  (  θ      /    2  ). It is standard that   cos  ⁡  θ  =            1      −              t        2                    1      +              t        2             and   sin  ⁡  θ  =            2      t              1      +              t        2            . So our equation becomes            2              t        2                    1      +              t        2              ⋅            (      1      −      t              )        2                    1      +              t        2              =  1  +  y  .Take the square root, and clear denominators. We get      2    t  (  1  −  t  )  =      1    +    y    (  1  +      t    2    )  .This is a quadratic in t.

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