Find the exact value of the trigonometric function given that sin(u) = -12/13 and cos(v) = -21/29. (Both u and v are in Quadrant III.)

Answer 1

One of the important concepts here is to realize that we can use the terminal angles to determine the values of various trig ratios. Here is a sketch of these problems:

We know that angle u is actually between 180° and 270° but we can use the angle u' which is the terminal angle and we can create a triangle as shown in red with a side that is -12 (that's right, negative 12) even though a negative side doesn't make sense in the real world. We can use the Pythagorean Theorem to solve for the top side of this triangle and get 5 but we must also realize that, because it's in Q3 it is actually a -5. Thusly, cos(u) = cos(u') = -5/13, tan(u) = tan(u') = -12/-5 = 12/5, csc(u) = csc(u') = 13/-12, sec(u) = sec(u') = -13/5, cot(u) = cot(u') = -5/-12 = 5/12.

We can do the same with v.

If your question is regarding double angles or half angles, we can use the double angle or half angle identities along with these basic trig values to find their exact values.