Solve for x sec(x)<0

Problem

sec(x)<0

Solution

1. Relate to cosine by using the reciprocal identity sec(x)=1/cos(x)

2. Determine the sign of the expression. Since the numerator 1 is positive, the fraction 1/cos(x) is negative only when the denominator is negative.

3. Identify the condition for the inequality to hold, which is cos(x)<0

4. Find the intervals on the unit circle where the xcoordinate (cosine) is negative. This occurs in the second and third quadrants.

5. Express the solution in terms of x For a single rotation [0,2*π) the condition is met when π/2<x<(3*π)/2

6. Generalize the solution by adding multiples of the period 2*π*n where n is any integer.

Final Answer

sec(x)<0⇒π/2+2*π*n<x<(3*π)/2+2*π*n

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