3 sin ⁡ x + 4 cos ⁡ y = 5, 4 sin ⁡ y + 3 cos ⁡ x = 2 How to find sin ⁡ x, sin ⁡ y, cos ⁡ x, cos ⁡ y

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3  sin  ⁡  x  +  4  cos  ⁡  y  =  5,   4  sin  ⁡  y  +  3  cos  ⁡  x  =  2 How to find   sin  ⁡  x,   sin  ⁡  y,   cos  ⁡  x,   cos  ⁡  y

levraijournalk1o · Accepted Answer

We have   4  cos  ⁡  y  =  5  −  3  sin  ⁡  x. Similarly we have   4  sin  ⁡  y  =  2  −  3  cos  ⁡  x. Squaring and add:  16  =  (  5  −  3  sin  ⁡  x      )    2    +  (  2  −  3  cos  ⁡  x      )    2    =  38  −  30  sin  ⁡  x  −  12  cos  ⁡  xso   15  sin  ⁡  x  +  6  cos  ⁡  x  =  11 which gives you possible   sin  ⁡  x  ,  cos  ⁡  x (i.e., solve the system   15  sin  ⁡  x  +  6  cos  ⁡  x  =  11  ,      sin    2    ⁡  x  +      cos    2    ⁡  x  =  1 to get   sin  ⁡  x  =            55      ±      4              35              87  , and similar for   cos  ⁡  x, or solve for   tan  ⁡      1    2    x  =            15      ±      2              35              17  ) and hence   sin  ⁡  y  ,  cos  ⁡  y

Kareem Mejia · Accepted Answer

Hint: starting with   sin  ⁡  x  cos  ⁡  y  +  cos  ⁡  x  sin  ⁡  y  =      1    6   and using that   cos  ⁡  y  =            5      −      3      sin      ⁡      x        4  ,   sin  ⁡  y  =            2      −      3      cos      ⁡      x        4   we can obtain  sin  ⁡  x  ⋅  (  5  −  3  sin  ⁡  x  )  +  cos  ⁡  x  ⋅  (  2  −  3  cos  ⁡  x  )  =      2    3    5  sin  ⁡  x  +  2  cos  ⁡  x  =      11    3      ⟹    cos  ⁡  x  =      11    6    −  2.5  sin  ⁡  xWe also have that       cos    2    ⁡  x  +      sin    2    ⁡  x  =  1Substitute   cos  ⁡  x and solve quadratic to find   sin  ⁡  x

3 sin x+4 cos y=5, 4 sin y+3 cos x=2 How to find sin x, sin y, cos x, cos y

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