Explanation:
sin(theta) and cos(theta) are both negative.
that means the angle is in the 3rd quadrant (bottom left) - between 180° and 270°.
doubling the angle brings the result between 2×180 = 360° = 0° and 2×270 = 540° = 180°.
so, sine is positive.
the double angle formulas are
sin(2×theta) = 2×sin(theta)×cos(theta)
cos(2×theta) = cos²(theta) - sin²(theta) =
= 2×cos²(theta) - 1 =
= 1 - 2×sin²(theta)
cos(2×theta) = 1 - 2×sin²(theta) = 1 - 2×(-35/37)² =
= 1 - 2×1225/1369 = 1 - 2450/1369 =
= 1369/1369 - 2450/1369 =
= -1081/1369
sin(2×theta) = 2×sin(theta)×cos(theta) =
= 2×(-35/37)×cos(theta)
cos²(theta) + sin²(theta) = 1
cos²(theta) = 1 - sin²(theta)
cos(theta) = sqrt(1 - sin²(theta)) =
= sqrt(1 - (-35/37)²) =
= sqrt(1 - 1225/1369) =
= sqrt(144/1369) = ±12/37
as cos(theta) < 0, it is -12/37
sin(2×theta) = 2×(-35/37)×cos(theta) =
= 2×(-35/37)×(-12/37) = 840/1369
FYI :
theta = 251.0753556...°
2×theta = 142.1507112...° (502.1507112... - 360)