If cos >0 and sin <0 what quadrant is it in?

Consider the rule #C-A-S-T# or All Slow Turtles Crawl for this type of problem.

Both of these rules indicate in which quadrant the trigonometric function is positive. In all quadrants except the first, only 1 out of the three trigonometric functions are positive; the other two are negative.

Quadrant 1: All are positive
Quadrant 2: Sine is positive
Quadrant 3: Tangent is positive
Quadrant 4: Cosine is positive

The acronym and the expression mentioned above are meant to facilitate your ability to remember these. Beware, though that C-A-S-T starts in the 4th quadrant and then goes to the 1st, 2nd and finally the 3rd, while "All Slow Turtles Crawl# goes from 1st to 4th.

Now, back to the problem at hand.

If cosine is positive, then we are already limited to 2 quadrants: IV and I. However, in quadrant I, all the functions are positive, and the problem says that sin is negative in this case. This leaves us one option: quadrant IV.

Hopefully this helps!

Cosine correlates with values of #x#.
Values of #x# are positive in #"Quadrant I"# and #"Quadrant IV"#.
Cosine is positive in the same quadrants.

Sine correlates with values of #y#.
Values of #y# are negative in #"Quadrant III"# and #"Quadrant IV"#.
Sine is negative in the same quadrants.

The only quadrant where #x# is positive, so #cos(x)>0#, and #y# is negative, so #sin(x)<0#, is #"Quadrant IV"#.

An example of an angle in #"Quadrant 4"# is #(7pi)/4#.

#cos((7pi)/4)=sqrt2/2>0#

#sin((7pi)/4)=-sqrt2/2<0#