4 Answers

cot(pi/6) = cos(pi/6) / sin(pi/6) = (√3 / 2) / (1/2) = √3

Draw a pi/2, pi/3, pi/6 triangle. In the old days we called it 306090. Label the sides lengths 1, 2, root 3. small side opposite small angle, of course.
Identifiy the the pi/6 angle and write the ratio of the length of the sides adjacent to opposite.

those are in step with values which you will desire to understand, like pi/6 = 30 degree., cosecant is a million/sin, sin 30 is a million/2 cosecant 30 = 2 2^2 = 4 cotangent a million/tan tan 30 = a million/sqrt 3 cot 30 = sqrt 3 – (sqrt 3)^2= 3 those are exact values. i’d desire to think of for the pi/18 Cot tan pi/18 = cot 10 stages = tan 80 stages tan(80)= 2 tan (40)/a milliontan^2 (40) tan (40) = 2 tan(20)/ a million tan^2 (20) tan (20)= 2 tan (10)/ a million tan^2 (10) Substitutuing, tan (40) = 4 (tan (10)/a milliontan^2(10) / a million – (2 tan(10) / a milliontan^2(10))^2 tan(80) = 8 (tan (10)/a milliontan^2(10) / a million – (2 tan(10) / a milliontan^2(10))^2/ (a million (4 (tan (10)/a milliontan^2(10) / a million – (2 tan(10) / a milliontan^2(10))^2)^2 for ease of calculating permit T = tan(10) Then substituting lower back tan(80)= 8 T/a millionT^2 / a million (2T /a millionT^2)^2 / a million (4 T/a millionT^2/a millionT^2/a million(2T/a millionT^2)^2 Now replace on the left a million/T for tan(80) and remedy for T that would desire to be an exact value.

pi/6 equals 30 degree
cotg (30) = 1/(tg 30) = 1/ (sqrt (3) / 3) = sqrt (3) = 1.732050808