Solve for x cos(2x)=-1/2

Problem

cos(2*x)=−1/2

Solution

1. Identify the general solution for the cosine function. For an equation cos(θ)=a the solution is θ=±arccos(a)+2*π*k where k is any integer.

2. Determine the principal value of arccos(−1/2) Since cos((2*π)/3)=−1/2 the reference angle is (2*π)/3

3. Set up the equations for the argument 2*x based on the general solution.

2*x=(2*π)/3+2*π*k

2*x=−(2*π)/3+2*π*k

4. Isolate x by dividing both sides of each equation by 2

x=π/3+π*k

x=−π/3+π*k

Final Answer

x=±π/3+π*k,k∈ℤ

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