sec(arctan(-3/5))
Let's work with the first part: arctan(-3/5)
The domain for arctan is limited to the first and fourth quadrants. Because the lengths are negative, the triangle must lie in the 4th quadrant.
tangent corresponds to (opp)/(adj):
color(white)(--)5
color(white)(.)color(black)(----)
color(white)(x)color(black)(\\)color(white)(--......)|
color(white)(x)color(white)(.)color(black)(\\)color(white)(---)|
color(white)(x)color(white)(..)color(black)(\\)color(white)(-.......)|
color(black)(x)color(white)(....)color(black)(\\)color(white)(-.... .)|
color(white)(x)color(white)(.....)color(black)(\\)color(white)(-....)|-3
color(white)(x)color(white)(.......)color(black)(\\)color(white)(-..)|
color(white)(x)color(white)(........)color(black)(\\)color(white)(-)|
color(white)(x)color(white)(..........)color(black)(\\)color(white)(..)|
color(white)(x)color(white)(...........)color(black)(\\)color(white)(.)|
color(white)(x)color(white)(.............)color(black)(\\)color(white)()|
We have this picture. Using it, we need to find sec. To do that, we need (hyp)/(adj).
adj is 5, but to find the hyp, we need to use pythagorean's theorem: sqrt((-3)^+5^2)=sqrt(34).
Now we have all our info: sec=(hyp)/(adj)=(sqrt(34))/5
