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

Problem

cos(2*x)=−1/2

Solution

1. Identify the reference angle for the cosine function. We know that cos(θ)=1/2 when θ=π/3

2. Determine the quadrants where the cosine is negative. Cosine is negative in the second and third quadrants.

3. Find the general solutions for the argument 2*x by applying the quadrant rules π−θ and π+θ

2*x=π−π/3+2*k*π

2*x=π+π/3+2*k*π

4. Simplify the expressions for 2*x

2*x=(2*π)/3+2*k*π

2*x=(4*π)/3+2*k*π

5. Solve for x by dividing all terms by 2 where k is any integer.

x=π/3+k*π

x=(2*π)/3+k*π

Final Answer

cos(2*x)=−1/2⇒x=π/3+k*π,(2*π)/3+k*π

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