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?
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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)
To determine in which quadrant angle x lies, we need to consider the signs of cosine and sine functions within each quadrant.
1. The given equation is cos 2x = 8/17. Since the cosine function is positive in the fourth quadrant (0 < x < π/2), we can infer that x lies in the fourth quadrant.
2. To find the measure of x in radians, we can use a trigonometric identity. We know that cos 2x = cos^2 x - sin^2 x. Substituting the given value of cos 2x = 8/17, we get:
cos^2 x - sin^2 x = 8/17
Now, we can apply the Pythagorean identity sin^2 x + cos^2 x = 1 to solve for sin^2 x:
1 - sin^2 x - sin^2 x = 8/17
2 sin^2 x = 9/17
sin^2 x = 9/34
Taking the square root of both sides, we find:
sin x = ± √(9/34)
Since x is in the fourth quadrant, x will be negative. Therefore, sin x = - √(9/34).
To find the measure of x in radians, we can use the inverse sine function:
x = arcsin(- √(9/34))
Now, using a calculator in radian mode, we can calculate the approximate value of x.