Which quadrant does θ lie, if sin θ > 0 and sec θ < 0

Which quadrant does θ lie, if

1) sin θ > 0 and sec θ < 0

2) cot θ < 0 and sin θ< 0

1) 2nd
2) 4th

EXPLANATION:

Ok, here's a picture of the quadrants of a simple graph. (http://www.algebra-by-example.com/images…

Now. Let's look at 1.
sin is known as y over radius (r). So that means, if sin θ is greater than 0, then it is either in the first or second quadrant. Now, sec is the opposite of cos, so if cos is x over r, that means that sec is r over x. We don't care about the radius. If sec θ (think x) is less than 0, that means it lies in either the second or the third quadrant. Combine the two statements, and the second quadrant is the correct one.

Now, let's look at 2.
Cot is the opposite of tan. Tan is y over x, so that means cot is x over y. Now, use logic. In any situation (whether it is cotangent or tangent), when both x and y are BOTH positive, they will lie in quadrant one. If both x and y are BOTH negative, they will lie in quadrant three (negative over a negative becomes a positive). That means, if either x ot y are negative (depending on the situation), cot or tan will lie either on the second or fourth quadrant.
Knowing that cot is less than 0, meaning that it is negative, either x or y are negative. That means it will lie in either the second or fourth quadrant.
Now, we already went over sin. Sin looks mainly at the y axis. Since sin θ here is less than 0, it is negative, meaning it lies either in the third or fourth quadrant.
Combine all statements and only the fourth quadrant fits.

1)
sin θ > 0 in quadrants 2 and 3

sec θ < 0 in quadrants 1 and 2

common quadrant is 2. So θ lies in quadrant 2.

2)
cot θ < 0 in quadrants 2 and 4

sin θ< 0 in quadrants 3 and 4

common quadrant is 4. So θ lies in quadrant 4.

The easiest way to do this is to realize that if secθ < 0, cosθ will be too.