solve the equation 4cos^2x-1=0
4c0s^2x - 1 = 0.
4cos^2x = 1,
Divide both sids by 4:
cos^2x = 0.25,
Take sqrt of both sides:
cosx = 0.5,
X = 60 deg.
To solve the equation 4cos^2(x) - 1 = 0, you can follow these steps:
Step 1: Rewrite cos^2(x) as (cos(x))^2.
The equation becomes 4(cos(x))^2 - 1 = 0.
Step 2: Move the constant term to the right side of the equation.
4(cos(x))^2 = 1.
Step 3: Divide both sides of the equation by 4.
(cos(x))^2 = 1/4.
Step 4: Take the square root of both sides of the equation.
cos(x) = ±√(1/4).
Step 5: Simplify the right side of the equation.
cos(x) = ±1/2.
Step 6: Use the inverse cosine (arccos) function to find the solutions for x.
The solutions for x will be the angles whose cosine is ±1/2.
Step 7: Determine the reference angle and the values for x using the unit circle.
The reference angle for cos(x) = ±1/2 is 60 degrees or π/3 radians. On the unit circle, the cosine is positive in the first and fourth quadrants and negative in the second and third quadrants.
Therefore, the solutions for x are:
x = arccos(1/2) = 60 degrees + 360n degrees or π/3 + 2πn radians (where n is an integer)
x = arccos(-1/2) = 180 - 60 degrees + 360n degrees or π - π/3 + 2πn radians (where n is an integer)
These are the values of x that satisfy the equation 4cos^2(x) - 1 = 0.