Solve the following equation for − 180 ∘ ≤ x ≤ 180 ∘ sin ⁡ 2 x = tan ⁡ x .I am aware that I can convert sin ⁡ 2 x = 2 sin ⁡ x cos ⁡ x. However, I am not sure where to go from here to find the values for x that will make the equation hold. Any help with solving this would be very much appreciated.

Question

Solve the following equation for   −      180    ∘    ≤  x  ≤      180    ∘    sin  ⁡  2  x  =  tan  ⁡  x  .I am aware that I can convert   sin  ⁡  2  x  =  2  sin  ⁡  x  cos  ⁡  x. However, I am not sure where to go from here to find the values for x that will make the equation hold. Any help with solving this would be very much appreciated.

Janiyah Patton · Accepted Answer

You have the correct approach.  2  sin  ⁡      x    cos  ⁡      x    =                    sin        ⁡                  x                            cos        ⁡                  x                      ⇒  sin  ⁡      x    (  2      cos    2    ⁡      x    −  1  )  =  0  ⇒  sin  ⁡      x    cos  ⁡      2    x    =  0Now either   sin  ⁡      x    =  0 or   cos  ⁡      2    x    =  0, for the first search all the solutions in   [  −  π  ,  π  ] and for second   −  π  ≤  x  ≤  π  ⇒  −  2  π  ≤  2  x  ≤  2  π. Thus first gives   x  =  0  ,  π  ,  −  π and second gives   2  x  =  ±      π    2    ,  ±            3      π        2  , thus   x  =  ±      π    4    ,  ±            3      π        4  . Thus solution set is   {  0  ,  ±  π  ,  ±      π    4    ,  ±            3      π        4    }

Kyle Sutton · Accepted Answer

hint  sin  ⁡  (  2  x  )  −  tan  ⁡  (  x  )  =  sin  ⁡  (  x  )  (  2  cos  ⁡  (  x  )  −      1          cos      ⁡      (      x      )        )  =  tan  ⁡  (  x  )  cos  ⁡  (  2  x  )  =  0  x  ∈  {  −  180  ,  −  135  ,  −  45  ,  0  ,  45  ,  135  ,  180  }

Solve the following equation for − 180 ∘ </msup>

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