Look:

You have a right triangle, with sides of length 3, 1, and . Surely you can evaluate the trigonometric functions here (if not, start paying attention!). Just remember that these lengths are "directed," so you need to take into account the negative signs. For example,

Although for our angle , , the angle made with thepositive-axisis in quadrant III, so the cosine is negative and we have .

You should be able to do the rest in similar fashion.

If the pulley is rotating at , then a point on the edge of the pulley will be traveling at , where is the radius of the pulley in feet. So we have .

These are the quadrants, along with the signs of the trigonometric functions in each:

A mnemonic that is sometimes used for the signs is:

All - All three (sin, cos, and tan) are positive in QI

Students - Only sin is positive in QII

Take - Only tan is positive in QIII

Calculus - Only cos is positive in QIV

Another way of looking at it is to use the unit circle:

By drawing a perpendicular to the -axis, you can see that every point on the unit circle is of the form , where is the angle between the positive -axis and the radial line containing the point. Thus you can easily see that the sine is positive wherever is positive (quadrants I and II), and the cosine is positive wherever is positive (quadrants I and IV).