If the tangent of an angle is 1/6, there are a few ways to find the other trig values. First, cot(alpha) = 1/tan(alpha) so 1 over 1/6 is 6. The formula sin2(alpha) + cos2(alpha)=1 can be divided through by cos2 to give tan2(alpha) + 1= sec2(alpha). The problem with plugging the 1/6 into that is you do not know which part of the circle or graph it is in so sec(alpha) upon solving would be sqrt[1/36 + 1] = sqrt[37/36] = sqrt(37)/6. It might be negative depending on the original angle but let's just assume it is positive for now. That brings us to sin and cos. Since sec(alpha) = 1/cos(alpha), it = the reciprocal of sqrt(37)/6 which is 6*sqrt(37)/37. Finally plug that back into sin2(alpha) + cos2(alpha)=1 to solve for cos(alpha).
To summarize, use cot(alpha) = 1/tan(alpha) to get cot(alpha). Use sec(alpha)= sqrt(tan2(alpha)+1) and then use that to get cos(alpha) since cos(alpha) = 1/sec(alpha). Finally, use sin2(alpha) + cos2(alpha)=1 to get sin(alpha). Since tan is positive in both quadrants 1 and 3, the sign on some of the trig functions might differ.
