help for calculus final?

Question

can anyone explain reference triangles for trig functions
example:cos(tan^-1(x/3))
I know you have to use trig and the 4 quadrants drawing some kind of triangle and labeling sides but I need help!

um 1st guy...not a very helpful answer

Answer 1

The first guy was a spambot, not a real person. Anyway, the first thing to do is note that since the range of the arctan function is (-π/2, π/2) and the cosines of all those angles are positive, we have cos (arctan (x/3)) = √(cos² (arctan (x/3))). After this, it's just basic algebra, plus one use of a trig identity (namely, sec² θ = 1 + tan² θ):

√(cos² (arctan (x/3)))
√(1/sec² (arctan (x/3)))
√(1/(1 + tan² (arctan (x/3))))
√(1/(1 + (x/3)²))
√(1/(1 + x²/9))
√(9/(9 + x²))
3/√(9+x²)

And we are done.