Mark any solutions to the equation 2cos2x - 1 = 0. The answer can be more than one. Thank you
π/8
-7π/4
15π/4
3π/4
2 cos (2x) - 1 = 0
cos (2x) = 1/2
2x = π/3 or 2x = 2π-π/3 or 5π/3
x = π/6 or x = 5π/6
none of the given answers apply
I will assume you meant
2 cos^2 x - 1 = 0
by definition:
2cos^2 x -1 = cos (2x)
so cos(2x) = 0
2x = π/2 or 2x = 3π/2
x = π/4 or x = 3π/4
I see one of those answers. 3π/4
looking at the others, note that -7π/4 is coterminal with π/4
so it would be a solution as well
To find the solutions to the equation 2cos(2x) - 1 = 0, we need to isolate the cosine term and then solve for x.
Step 1: Add 1 to both sides of the equation:
2cos(2x) = 1
Step 2: Divide both sides of the equation by 2:
cos(2x) = 1/2
Now, we need to find the values of 2x where cos(2x) is equal to 1/2.
Step 3: Take the inverse cosine (also known as arccos) of both sides to find the angle(s):
2x = arccos(1/2)
Step 4: Use the unit circle or a calculator to find the angles where cos(x) = 1/2. These angles are the solutions to the equation.
From the unit circle, we know that the angles where cos(x) = 1/2 are π/3 and 5π/3 (in the range 0 to 2π). However, since we have 2x instead of x, we need to divide these angles by 2.
Step 5: Divide the angles by 2:
x = π/3 / 2 = π/6
and
x = 5π/3 / 2 = 5π/6
Therefore, the solutions to the equation 2cos(2x) - 1 = 0 are:
x = π/6 and x = 5π/6
Note: None of the solutions you provided (π/8, -7π/4, 15π/4, 3π/4) are correct.
To find the solutions to the equation 2cos(2x) - 1 = 0, we need to isolate the cosine term and then find the values of x that give us a cosine value that satisfies the equation.
1. Start with the original equation: 2cos(2x) - 1 = 0.
2. Add 1 to both sides of the equation to isolate the cosine term: 2cos(2x) = 1.
3. Divide both sides of the equation by 2 to further isolate the cosine term: cos(2x) = 1/2.
4. Now, we need to find the angles whose cosine value is equal to 1/2. We can use the unit circle or the inverse cosine function (arccos) to find these angles.
- The cosine of an angle gives the x-coordinate of the point on the unit circle corresponding to that angle.
- On the unit circle, the x-coordinate is positive in the first and fourth quadrants. So, we need to find the angles in these quadrants that have a cosine value of 1/2.
5. The angle in the first quadrant with a cosine of 1/2 is π/3 (or 60 degrees).
6. To find the angle in the fourth quadrant, we can use the symmetry of the unit circle. Since cosine is an even function, the cosine value is the same in the fourth quadrant as it is in the first quadrant.
- That means the angle in the fourth quadrant with a cosine of 1/2 is the supplement of the angle in the first quadrant.
- The supplement of π/3 is 2π - π/3 = 5π/3.
7. So, the solutions to the equation 2cos(2x) - 1 = 0 are x = π/6 and x = 5π/6.
However, the solutions you provided (π/8, -7π/4, 15π/4, 3π/4) do not match the solutions found from the equation.