At first, one should appropriately define the system of equations. Instead of machine precission numbers we prefer exact numbers therefore we would define:
{a, b} /. Solve[{ 9/5 a + b == 1/100, 2 a + b == 0}, {a, b}]
{{-(1/20), 1/10}}
Now
pts = {Sin[u], Sin[2 u]} /. Solve[{ x == Sin[u],
Sin[2 u] == -(1/20) x + 1/10, -2 Pi <= u <= 2 Pi},
{u}, {x}, Reals]
{ {Sin[2 ArcTan[Root[1 - 41 #1 + 2 #1^2 + 39 #1^3 + #1^4 &, 1]]],
Sin[2 (2 Pi + 2 ArcTan[ Root[1 - 41 #1 + 2 #1^2 + 39 #1^3 + #1^4 &, 1]])]},
{Sin[ 2 ArcTan[Root[1 - 41 #1 + 2 #1^2 + 39 #1^3 + #1^4 &, 2]]],
Sin[2 (2 Pi + 2 ArcTan[ Root[1 - 41 #1 + 2 #1^2 + 39 #1^3 + #1^4 &, 2]])]},
{Sin[2 ArcTan[Root[1 - 41 #1 + 2 #1^2 + 39 #1^3 + #1^4 &, 3]]],
Sin[4 ArcTan[Root[1 - 41 #1 + 2 #1^2 + 39 #1^3 + #1^4 &, 3]]]},
{Sin[2 ArcTan[Root[1 - 41 #1 + 2 #1^2 + 39 #1^3 + #1^4 &, 4]]],
Sin[4 ArcTan[Root[1 - 41 #1 + 2 #1^2 + 39 #1^3 + #1^4 &, 4]]]} }
N @ %
{{-0.0513515, 0.102568}, {-0.997173, 0.149859},
{0.0488373, 0.0975581}, {0.999687, 0.0500156}}
ParametricPlot[{ {Sin[u], Sin[2 u]},
{u, (-(1/20) u + 1/10) ConditionalExpression[1, -5/4 <= u <= 5/4]}},
{u, -2 Pi, 2 Pi},
Epilog -> {Red, PointSize[0.017], Point[pts]}]

Edit
Your original system works as well if you add the specification domain: Reals
(mind different variables here).
Solve[ a ({1.8, .01} - {2, 0}) + {2, 0} == {Sin[u], Sin[2 u]} &&
0 <= u < 2 Pi && a > 0, {u, a}, Reals]
Solve::ratnz: Solve was unable to solve the system with inexact coefficients.
The answer was obtained by solving a corresponding exact system
and numericizing the result. >>
{{u -> 0.0488568, a -> 9.75581}, {u -> 1.54578, a -> 5.00156},
{u -> 3.19297, a -> 10.2568}, {u -> 4.63718, a -> 14.9859}}
alternatively substituting {1.8, .01}
by {9/5, 1/100}
yields appropriate results without adding domain specification, however in terms or radicals not explicitly real.
Even though 0 <= u < 2 Pi
and a > 0
inequalities should restrict variables u
and a
to the real numbers, nonetheless most likely on the internal processing level there might appear some inconsistency. Therefore one should explicitly add the domain specification.