Trouble solving a trigonometric equationI have this problem in the book and no matter how many times I try I cannot isolate θ: (displayed) 2 = ( θ − 2 c o s θ ) ( c o s θ )

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Trouble solving a trigonometric equationI have this problem in the book and no matter how many times I try I cannot isolate θ:                    (displayed)                    2        =        (                  θ                −        2        c        o        s                  θ                )        (        c        o        s                  θ                )

Frances Dodson · Accepted Answer

You can&#039;t isolate it.Just like solving   x      e    x    =  a for   x in terms of   a required creating a new function (Lambert W), this can only be solved numerically.I can see by looking at the equation that there are an infinite number of solutions (look near where   cos  ⁡  (  θ  )  =  0). Note that, once   θ  &amp;gt;  2,   θ  −  2  cos  ⁡  θ  &amp;gt;  0, so all the roots there are supplied by the   cos  ⁡  θ term. Since   cos  ⁡  θ  =  0 implies that   θ  =  π  (  n  +      1    2    ) for some integer   n, you can get an estimate for the root near this by letting       r    n    =  x  +  π  (  n  +      1    2    ) any applying one Newton&#039;s iteration to get a better approximation.For looking around   θ  =  0, let   cos  ⁡  θ  ∼  1  −            θ      2        2   and try to get an approximation for   θSo plot the function and see where the roots are.

Trouble solving a trigonometric equation I have this problem in the book and no matter how many times I try I cannot isolate θ: 2 = (θ - 2cos theta)(cos theta)

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