Answer
The other ratios:
cos: $\frac{6}{\sqrt 61}$
sin: $\frac{5}{\sqrt 61}$
sec: $\frac{\sqrt 61}{6}$
csc: $\frac{\sqrt 61}{5}$
cot: $\frac{6}{5}$
Work Step by Step
Remember the acronym, "SOHCAHTOA"
$sine=\frac{opp}{hyp}$
$cos=\frac{adj}{hyp}$
$tan=\frac{opp}{adj}$
And the opposites of these functions switch everything.
Since we are given $tan=\frac{5}{6}$, we know that our opposite is 5 and our adjacent is 6. From there we can calculate a hypotenuse of $\sqrt 61$ (pythagorean theorem: $5^2 + 6^2 = c^2$
From there, you can find the remaining ratios.
