Question

if cos theta < 0 and cot theta > 0 then the terminal point determined by theta is in what quadrant?

Answer 1

If Cosine is positive, the terminal point is to the right of the y axis with would put it in either Quadrant 1 (Q1) or Quadrant 4 (Q4)
If Cosine is negative the terminal point is to the left of the y axis which would put it in either Quadrant 2 (Q2) or Quadrant 3 (Q3).

If Sine is positive, the terminal point is above the x axis which would put it in either Q1 or Q2
If Sine is negative, the terminal point is below the x axis which would put it in either Q3 or Q4.

If cosine is negative, then the terminal point is in Q2 or Q3
Since the cotangent is positive and the cotangent equals the cosine divided by the sine, the sine must have the same sign as the cosine. Therefore the sine is also negative.
If sine is negative, then the terminal point is in Q3 or Q4.

Therefore, the terminal point must be in Q3 since that is the quadrant in which both sine and cosine are negative.

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Answer 2

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