2cosx=√2+√2+2cos4x
not sure just how sloppy you are being with the parentheses, You might mean
√2+√(2+2cos4x)
or
√(2+√(2+2cos4x))
either way it's kinda sloppy
first step is probably to see what cos4x reduces to
2cosx
2cosx
√2+√2+2cos4x=2cosx
To solve the equation 2cos(x) = √2 + √2 + 2cos(4x), we can follow the steps below:
Step 1: Simplify the equation by combining like terms:
2cos(x) = 2√2 + 2cos(4x)
Step 2: Divide both sides of the equation by 2:
cos(x) = √2 + cos(4x)
Step 3: Use the trigonometric identity cos(2A) = 2cos^2(A) - 1 to replace cos(4x) with cos(2(2x)):
cos(x) = √2 + cos^2(2x) - 1
Step 4: Rearrange the equation to get the cosine function on one side:
cos(x) - cos^2(2x) = √2 - 1
Now, solving this equation algebraically may be a bit challenging. We can utilize a graphical or numerical approach to find the solution.
Graphical Approach:
Step 5: Graph both sides of the equation separately on the same coordinate system.
On the left side of the equation (cos(x) - cos^2(2x)), we will have a combination of a cosine function and a cosine squared function.
On the right side of the equation (√2 - 1), we will have a constant value.
Step 6: Find the x-coordinates where the two graphs intersect. These points will represent the solutions to the equation.
Numerical Approach:
Step 5: Use a numerical method such as the Newton-Raphson method or the bisection method to approximate the values of x that satisfy the equation.
With either approach, you should be able to find the values of x that make the equation true.