Find all solutions to tan ⁡ ( x − π ) + 2 sin ⁡ ( x − π ) = 0 on [ 0 , 2 π )

Question

Find all solutions to   tan  ⁡  (  x  −  π  )  +  2  sin  ⁡  (  x  −  π  )  =  0 on   [  0  ,  2  π  )

ysik92ASw · Accepted Answer

Given  tan  ⁡  (  x  −  π  )  +  2  sin  ⁡  (  x  −  π  )  =  0We know that            sin      ⁡      (      x      −      π      )              cos      ⁡      (      x      −      π      )        +  2  sin  ⁡  (  x  −  π  )  =  0    sin  ⁡  (  x  −  π  )  [      1          cos      ⁡      (      x      −      π      )        +  2  ]  =  0    If   sin  ⁡  (  x  −  π  )  =  0    x  −  π  =      sin          −      1        ⁡  (  0  )  =  0    x  =  πIf       1          cos      ⁡      (      x      −      π      )        +  2  =  0        1          cos      ⁡      (      x      −      π      )        =  −  2    cos  ⁡  (  x  −  π  )  =      1          −      2          x  −  π  =      cos          −      1        ⁡  (      1          −      2        )    x  −  π  =      60    ∘      x  =      240    ∘      x  =            4      π        3

Find all solutions to tan(x-pi)+2sin(x-pi)=0 on [0,2pi)

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