Question

If sin theta is negative and cot theta is positive, in which quadrant does theta terminate?

Answer 1

All you need to know is that:

• In the first quadrant, both coordinates are positive
• In the second quadrant, x is negative and y is positive
• In the third quadrant, both coordinates are negative
• In the fourth quadrant, x is positive and y is negative
• The sine is the y coordinate of the points on the unit circle
• The cosine is the x coordinate of the points on the unit circle
• The cotangent function is defined as the ration between the cosine and the sine of an angle.

So, if sin theta is negative and cot theta is positive, you're saying that you're looking at a point with negative y coordinate and such that

$\dfrac{\cos(\theta)}{\sin(\theta)}>0$

The ratio of two numbers is positive if and only if they have the same sign, so the cosine must by negative as well.

Saying that an angle has negative sine and cosine is the same as saying that the angle identifies a point on the unit circle where both x and y coordinates are negative.

So, if you look at the bullet list at the beginning, you'll see that the point is in the third quadrant.