Solve for x cos(x/2)=( square root of 2)/2

Problem

cos(x/2)=√(,2)/2

Solution

1. Identify the reference angle for the cosine function. The value √(,2)/2 corresponds to an angle of π/4 radians (or 45.

2. Determine the general solution for the cosine equation cos(θ)=a Since cosine is positive in the first and fourth quadrants, the solutions for the argument x/2 are:

x/2=π/4+2*k*π

x/2=−π/4+2*k*π

where k is any integer.

3. Isolate x by multiplying both sides of each equation by 2

x=2*(π/4+2*k*π)⇒x=π/2+4*k*π

x=2*(−π/4+2*k*π)⇒x=−π/2+4*k*π

4. Combine the results into a single expression using the ± notation.

Final Answer

cos(x/2)=√(,2)/2⇒x=±π/2+4*k*π

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