Solve for x sec(x)=-2

Problem

sec(x)=−2

Solution

1. Rewrite the equation using the reciprocal identity for the secant function.

cos(x)=1/sec(x)

2. Substitute the given value of the secant function into the identity.

cos(x)=−1/2

3. Identify the reference angle by finding the value of x in the first quadrant where the cosine is positive.

(x_ref)=arccos(1/2)=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 the period of the cosine function, 2*π*n where n is any integer.

x=(2*π)/3+2*π*n

x=(4*π)/3+2*π*n

Final Answer

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

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