Because when you plug those last two angles in the original equation, the equation is false.

The first two make the original equation true.

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6sinX = 5 -8cosX

Square both sides,

36sin^2(X) = 25 -80cosX +64cos^2(X)

36(1 -cos^2(X)) = 25 -80cosX +64cos^2(X)

100cos^2(X) -80cosX -11 = 0

Using the quadradtic formula,

cosX = 0.91962 or -0.11962

When cosX = 0.91962 .......positive cosine value, so X is in the 1st or 4th quadrants,

X = 23.1294 deg .....1st quadrant

X = 336.8706 deg ....4th quadrant

When cosX = -0.11962 ......negative cosine value, so X is in the 2nd or 3rd quadrants,

X = 96.8702 deg .....2nd quadrant

X = 263.1298 deg ....3rd quadrant

Since X = 23.1294 deg and X = 263.1298 deg do not make the original equation true, then it means X is not in the 1st and 3rd quadrants.