find, to the nearest degree, the solution set of 4 cos^2x-1=0 in the intervak between )degrees and 360degrees.
4 cos^2x-1=0
cos^2 x = 1/4
cosx = ± 1/2
In I x = 60
in II , x = 180-60 = 120
in III, x = 180+60 = 240
in IV , x = 360-60 = 300
To find the solution set of the equation 4cos^2(x) - 1 = 0 in the interval between 0 degrees and 360 degrees, you can follow these steps:
1. Start by isolating the cosine term on one side of the equation:
4cos^2(x) = 1
2. Divide both sides of the equation by 4:
cos^2(x) = 1/4
3. Take the square root of both sides of the equation:
cos(x) = ±√(1/4)
4. Simplify the right side of the equation:
cos(x) = ±1/2
5. Now, to find the values of x, you need to determine the angles in the interval between 0 and 360 degrees where the cosine function equals ±1/2. To do this, you can consult the unit circle or use trigonometric ratios.
- For cos(x) = 1/2:
You can find this value on the unit circle for angles 60 degrees and 300 degrees, as cos(60°) = cos(300°) = 1/2.
- For cos(x) = -1/2:
You can find this value on the unit circle for angles 120 degrees and 240 degrees, as cos(120°) = cos(240°) = -1/2.
6. Therefore, the solution set of the equation in the given interval is:
{60°, 120°, 240°, 300°}
Note that these values can be expressed in radians as well if needed.