How can I find x,y for: sin ⁡ x + sin ⁡ y = 0 sin ⁡ 2 x + sin ⁡ 2 y = 0I've tried to utilize the identity sin ⁡ 2 x = 2 sin ⁡ x cos ⁡ x but wasn't able to reach the solution.

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How can I find x,y for:  sin  ⁡  x  +  sin  ⁡  y  =  0    sin  ⁡  2  x  +  sin  ⁡  2  y  =  0I&#039;ve tried to utilize the identity   sin  ⁡  2  x  =  2  sin  ⁡  x  cos  ⁡  x but wasn&#039;t able to reach the solution.

lilao8x · Accepted Answer

You&#039;ve used the correct identity, just go on:  sin    x  =  −  sin    yhence the second equations is equivalent to  sin  ⁡  y  (  cos    y  −  cos    x  )  =  0and now you just split the solution into  sin    y  =  0      OR      (  cos    y  −  cos    x  )  =  0.In the first case, when   sin    y  =  0, then   sin    x  =  0 as well, so   (  x  ,  y  )  =  (  k  ;  n  )  π for   k  ,  n  ∈      N  In the second case one has to consider both conditions, i.e.   cos    y  =  cos    x implies x and y must be either equal or x=−y (symmetrical along horizontal axis + consider the period) from which the condition   sin    x  =  −  sin    y chooses the latter.

How can I find x,y for: sin ⁡ x + sin ⁡ y

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