Trig
posted by Jus .
The angle 2x lies in the fourth quadrant such that cos2x=8/17.
1.Which quadrant contains angle x?
2. What is the measure of x, in radians?

I know that angle 2x is in quadrant 4, but quadrant x...? I am not sure where it is and how to get the measure of it in radians.

Half of quadrant 4 is half of 270deg to 360 deg, or 135 to 190 deg, quadrant 2.

switch your calculator to radians
enter
8
÷
17
=
to get .470588...
press 2ndF, then cos
to get 1.08084
This would be the radian measure in the first quadrant, but you are in the fourth,
so 2pi  1.0804 = 5.20235
but this is 2x
so x = 5.20235/2
= 2.60117
(in degrees that would be 149.04º which is in compliance with what bobpursley told you above)
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