Question

2cos4x=1

Answer 1

2cos(4x)=1 cos4x=1/2 4x=+-arccos(4x)+2pi*n x=+-(1/4)*arccos(4x)+pi*n/2

Answer 2

To solve the equation 2cos(4x) = 1, we can use the inverse cosine function (arccos) to find the value of x.

First, let's isolate the cosine term:

2cos(4x) = 1

Divide both sides by 2:

cos(4x) = 1/2

Now, we can take the inverse cosine (arccos) of both sides to find the value of 4x:

4x = arccos(1/2)

The inverse cosine of 1/2 is 60 degrees or π/3 radians. However, keep in mind that the cosine function has a period of 2π (360 degrees). Therefore, we need to consider all possible solutions within one period.

Since cos(x) is positive in the first and fourth quadrants, we can add multiples of the period 2π to the initial solution.

To find the general solution, we divide both sides of 4x = π/3 by 4:

x = (π/3)/4

Simplifying, we get:

x = π/12

Adding multiples of 2π/4 to π/12, we can find the general solutions:

x = π/12 + (2π/4)n, where n is an integer.

So, the general solution to the equation 2cos(4x) = 1 is:

x = π/12 + (2π/4)n, where n is an integer.