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

Problem

cos(x)=−1/2

Solution

1. Identify the reference angle by finding the value of α such that cos(α)=1/2 in the first quadrant.

α=π/3

2. Determine the quadrants where the cosine function is negative. Since cos(x)=−1/2 x must be in Quadrant II or Quadrant III.

3. Calculate the solution in Quadrant II using the formula x=π−α

x=π−π/3=(2*π)/3

4. Calculate the solution in Quadrant III using the formula x=π+α

x=π+π/3=(4*π)/3

5. Generalize the solutions to include all coterminal angles by adding multiples of the period 2*π where n is any integer.

x=(2*π)/3+2*n*π

x=(4*π)/3+2*n*π

Final Answer

cos(x)=−1/2⇒x=(2*π)/3+2*n*π,(4*π)/3+2*n*π

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