:-

2 sin x = - √2

sin x = - 1 / √2 ______x in quadrants 3 and 4

x = 5π/4 , 7π/4

2sin x = -√2

sin x = -(√2)/2

x = arcsen -(√2)/2 = -45°

⁻¹8 sin x + 6√2 = 6 sin x + 5√2

8 sin x - 6 sin x = 5√2 - 6√2

2 sin x = -√2

sin x = -√2 / 2

x = sin⁻¹ (-√2 / 2)

x = 7π/4

8 * sin(x) + 6 * sqrt(2) = 6 * sin(x) + 5 * sqrt(2)

8 * sin(x) - 6 * sin(x) = 5 * sqrt(2) - 6 * sqrt(2)

2 * sin(x) = -sqrt(2)

sin(x) = -sqrt(2)/2

What was hard about that?  The values are on the unit circle.  They're principal values for sin(t)