Solve for x cos(2x)=-( square root of 3)/2

Problem

cos(2*x)=−√(,3)/2

Solution

1. Identify the reference angle for the cosine function. We know that cos(θ)=√(,3)/2 when the reference angle is π/6

2. Determine the quadrants where the cosine is negative. Since the value is −√(,3)/2 the angle 2*x must terminate in Quadrant II or Quadrant III.

3. Find the general solutions for the angle 2*x by applying the reference angle to the identified quadrants and adding the period 2*π*k where k is any integer.

2*x=π−π/6+2*π*k=(5*π)/6+2*π*k

2*x=π+π/6+2*π*k=(7*π)/6+2*π*k

4. Solve for x by dividing all terms in both equations by 2

x=(5*π)/12+π*k

x=(7*π)/12+π*k

Final Answer

x=(5*π)/12+π*k,(7*π)/12+π*k

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