sin^2 2θ − 5 sin 2θ − 1 = 0

To make the calculations easier, for now, replace "sin(2θ)" with x:

Then we have:     x² - 5x - 1  =  0

Using the quadratic formula:  x  =  [5 + √(25 + 4)] / 2  ≈  5.1926    

                                   or     x  =  [5 - √(25 + 4)] / 2  ≈  -0.19258

Since x = sin(2θ):       sin(2θ)  =  5.1926     or     sin(2θ)  =  -0.19258

     --->                           2θ  =  sin^-1(5.1928)     or     2θ  =  sin^-1(-0.19528)

     --->                             θ  =  sin^-1(5.1928) / 2     or   θ  =  sin^-1(-0.19528) / 2

Since sin does not have a value larger than 1, θ  =  sin^-1(5.1928) / 2  is an impossibility.

For  θ  =  sin^-1(-0.19528) / 2,  

      θ equals either 275.55°  or  354.4° (and all multiples of 180°, larger and smaller).