trigonometry equation, minus problem, sin ⁡ x + sin ⁡ 5 x = 0in the book , there are 2 solutions: x=60k , x=45+90k. in my solution, I got the first solution right, but the second solution with a minus. I don't understand what I did wrong. sin ⁡ ( x ) + sin ⁡ ( 5 x ) = 0 sin ⁡ ( x ) = − sin ⁡ ( 5 x ) sin ⁡ ( x ) = sin ⁡ ( − 5 x ) x = − 5 x + 360 k 6 x = 360 k x = 60 k x = 180 − ( − 5 x ) + 360 k x = 180 + 5 x + 360 k − 4 x = 180 + 360 k x = − 45 − 90 k

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trigonometry equation, minus problem,   sin  ⁡      x    +  sin  ⁡      5    x    =  0in the book , there are 2 solutions: x=60k , x=45+90k. in my solution, I got the first solution right, but the second solution with a minus. I don&#039;t understand what I did wrong.  sin  ⁡  (  x  )  +  sin  ⁡  (  5  x  )  =  0    sin  ⁡  (  x  )  =  −  sin  ⁡  (  5  x  )  sin  ⁡  (  x  )  =  sin  ⁡  (  −  5  x  )  x  =  −  5  x  +  360  k    6  x  =  360  k    x  =  60  k  x  =  180  −  (  −  5  x  )  +  360  k    x  =  180  +  5  x  +  360  k    −  4  x  =  180  +  360  k    x  =  −  45  −  90  k

Nathen Austin · Accepted Answer

All numbers of the form   45  +  90  k for integer k are also numbers of the form   −  45  −  90  m for integer m:  45  +  90  k  =  −  45  −  90  m      ⟺    90  (  k  +  1  )  =  90  (  −  m  )      ⟺    m  =  −  k  −  1  ,so your solution gives the same values as the book.

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