If theta is the measure of an angle in standard position with the given conditions, find the quadrant(s) in which the terminal side of theta lies.

a) cot theta < 0

b) cos theta > 0 , csc theta < 0

a) 2 and 4

b) cos+ in 1 and 4 but csc + in 1
so must be in 4

To determine the quadrant(s) in which the terminal side of an angle lies, you need to examine the signs of the trigonometric functions. Remember that in standard position, the initial side of the angle is the positive x-axis and the terminal side is the ray formed by rotating the initial side counter-clockwise.

a) For cot theta < 0:
Cotangent is the ratio of cosine to sine, so cot theta is negative when cosine is negative and sine is positive or when cosine is positive and sine is negative. In other words, the sign of cot theta depends on the signs of both cosine and sine.

To find the quadrants where cot theta < 0, you can split it into two scenarios:
1. Cosine is negative and sine is positive: In this case, theta would lie in the second quadrant where both x and y coordinates are negative.
2. Cosine is positive and sine is negative: In this case, theta would lie in the fourth quadrant where x is positive (cosine is positive) and y is negative (sine is negative).

Therefore, the terminal side of theta can lie in either the second quadrant or the fourth quadrant.

b) For cos theta > 0 and csc theta < 0:
Cosine is positive when theta lies in the first or fourth quadrant. From the given conditions, we know that cos theta is positive, so we can determine that theta lies in either the first or fourth quadrant.

Cosecant is the reciprocal of sine, so csc theta is negative when sine is negative. From the given condition that csc theta is negative, we can determine that sine is negative, which means theta lies in either the third or fourth quadrant.

Combining the information from the conditions, we find that theta must be in the fourth quadrant, where both cosine and sine are positive. Therefore, the terminal side of theta lies in the fourth quadrant.

In summary:
a) The terminal side of theta can lie in either the second quadrant or the fourth quadrant.
b) The terminal side of theta lies in the fourth quadrant.