Solve for x sec(x)=-2

Problem

sec(x)=−2

Solution

1. Rewrite the equation using the reciprocal identity sec(x)=1/cos(x)

1/cos(x)=−2

2. Solve for cos(x) by taking the reciprocal of both sides.

cos(x)=−1/2

3. Identify the reference angle by finding the value of θ such that cos(θ)=1/2

θ=π/3

4. Determine the quadrants where the cosine function is negative, which are Quadrant II and Quadrant III.

x=π−π/3=(2*π)/3

x=π+π/3=(4*π)/3

5. Generalize the solution 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

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

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