This is impossible to discuss without a picture, so please see the attached drawing.
The question tells us that the angle is in the 3rd quadrant, so it's between 180 and 270,
and laid out as in the drawing.
If you remember the definitions of the trig functions in terms of the sides of the
right triangle, you can read them straight off of the drawing:
sin(θ) = opp/hyp = -3/5
csc(θ) = hyp/opp = -5/3
cos(θ) = adj/hyp = -4/5
sec(θ) = hyp/adj = -5/4
tan(θ) = opp/adj = 3/4
cot(θ) = adj/opp = 4/3
To find the value of the angle, I can punch it up on my calculator,
but I have to look at the drawing and keep in mind that the whatever
the calculator says, the angle is that much past 180 degrees, so
I need to add 180 to it.
The arcsin (angle whose sin is) (3/5) is 36.869898 degrees
so the angle in the drawing is 180 plus that = 216.869898 degrees.
The question asks for the nearest second, so I have to convert the
decimal part of that angle to minutes and seconds. I'll just go off
by myself and do that.
The angle is 216 deg 52 min 11.63 sec, which rounds to 12 sec.
