Solve the equation 8 cos^2 X -2 =0 given that 180< or equal to X <or equal to 360 degrees.
8cos^2X - 2 = 0
8cos^2x = 2
cos^2x = 2/8 = 0.25
cosx = sqrt(0.25) = 0.5
Xr = 60o = Reference angle.
X-Xr = 180
x-60 = 180
X = 180+60 = 240o.
To solve the equation 8cos^2(x) - 2 = 0, we can use the following steps:
Step 1: Isolate the cosine term by moving the constant to the other side of the equation:
8cos^2(x) = 2
Step 2: Divide both sides of the equation by 8 to solve for the squared cosine term:
cos^2(x) = 2/8
Step 3: Simplify the right side of the equation:
cos^2(x) = 1/4
Step 4: Take the square root of both sides of the equation:
√[cos^2(x)] = ±√(1/4)
Since we are given that 180 ≤ x ≤ 360 degrees, we will only consider the positive square root.
Step 5: Solve for x by taking the inverse cosine (arccos) of both sides:
x = arccos(±√(1/4))
To find the value of x within the given range, we need to use a calculator or a table of trigonometric values. In this case, we are looking for the range where 180 ≤ x ≤ 360 degrees.
Step 6: Calculate the value of x within the given range:
x1 = arccos(√(1/4))
x2 = arccos(-√(1/4))
Using a calculator, we find that x1 is approximately 60 degrees and x2 is approximately 300 degrees.
Therefore, the solutions to the equation 8cos^2(x) - 2 = 0 within the given range of 180 ≤ x ≤ 360 degrees are x1 ≈ 60 degrees and x2 ≈ 300 degrees.