Determine the quadrant in which theta lies:

sec theta>0 and cot theta<0

sec theta is positive where cos theta is positive. That would be the first and fourth quadrants. Cot theta is negative where tan theta is negative, and that would be in the second and fourth quadrants. The only quadrant that satifies both criteria is the fourth.

because sec theta = 1/cos theta.

Both functions have the same sign

To determine the quadrant in which theta lies based on the given conditions, we need to analyze the signs of the trigonometric ratios sec theta and cot theta.

1. sec theta > 0:
The secant function is positive in either the first or the fourth quadrant.

2. cot theta < 0:
The cotangent function is negative in either the second or the fourth quadrant.

Combining these two conditions, we find that theta can only lie in the fourth quadrant, as it is the only quadrant where sec theta is positive and cot theta is negative.

Therefore, theta lies in the fourth quadrant.

why is sec theta positive where cos theta is positive?