Finding the general solution of a trigonometric equationFind the general solution of the equation 2 sin 2 ⁡ x + 5 sin ⁡ x cos ⁡ x − 8 cos 2 ⁡ x = − 2My working: 2 sin 2 ⁡ x + 5 sin ⁡ x cos ⁡ x − 8 cos 2 ⁡ x + 2 = 0 2 sin 2 ⁡ x + 5 sin ⁡ x cos ⁡ x + 8 sin 2 ⁡ x − 6 = 0 10 sin 2 ⁡ x + 5 sin ⁡ x cos ⁡ x − 6 = 0 5 − 5 cos ⁡ 2 x + 5 sin ⁡ x cos ⁡ x − 6 = 0I am giving my working for the sake of it...I know I'm getting nowhere. I really need a hint as to how I can take this forward.

Question

Finding the general solution of a trigonometric equationFind the general solution of the equation     2      sin    2    ⁡  x  +  5  sin  ⁡      x    cos  ⁡      x    −  8      cos    2    ⁡  x  =  −  2My working:     2      sin    2    ⁡  x  +  5  sin  ⁡      x    cos  ⁡      x    −  8      cos    2    ⁡  x  +  2  =  0     2      sin    2    ⁡  x  +  5  sin  ⁡      x    cos  ⁡      x    +  8      sin    2    ⁡  x  −  6  =  0     10      sin    2    ⁡  x  +  5  sin  ⁡      x    cos  ⁡      x    −  6  =  0     5  −  5  cos  ⁡  2  x  +  5  sin  ⁡      x    cos  ⁡      x    −  6  =  0I am giving my working for the sake of it...I know I&#039;m getting nowhere. I really need a hint as to how I can take this forward.

Zackary Mack · Accepted Answer

2      sin          2        ⁡      x    +  5  sin  ⁡      x    cos  ⁡      x    −  8      cos          2        ⁡      x    =  −  2      2      sin          2        ⁡      x    +  5  sin  ⁡      x    cos    ⁡          x        −  8      cos          2        ⁡      x    =  −  2  ⋅  1  =  −  2      (          sin              2              ⁡          x      +              cos                  2                    ⁡              x              )    4      sin          2        ⁡      x    +  5  sin  ⁡      x    cos  ⁡      x    −  6      cos          2        ⁡      x    =  0        (    d    i    v    i    d    e        b    o    t    h        t    o              cos              2              ⁡          x            )    4      tan          2        ⁡      x    +    5    tan    ⁡          x        −    6    =    0      tan  ⁡      x    =            −      5      ±      7        8    tan  ⁡      x    =  −      3    2    ⇒  x  =  −  arctan  ⁡            3      2        +  n  π  ,  n  ∈      Z      tan  ⁡      x    =      1    4    ⇒  x  =  arctan  ⁡            1      4        +  n  π  ,  n  ∈         Z

Finding the general solution of a trigonometric equation Find the general solution of the equation 2 sin^2x+5 sin (x) cos (x) -8 cos^2x=-2

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