Question

cot pi/3

Answer 1

Cotangent(pi/3) ; let's call cotangent cot
cot(pi/3) = 1/tan(pi/3)
convert pi/3 radians to degrees.
(pi/3) * (180/pi) = 60
1/tan(60) ; look at a reference triangle (in sources)
tangent of 60 = opposite/adjacent = (sqrt(3)/2) / (1/2) which is really sqrt(3)/2 * 2/1 (the reciprocal)
that is then equal to 2sqrt(3)/2
the 2's cancel out and the answer is:
tangent(60) = sqrt(3)
now the cot = 1/tan
cot(pi/3) = 1/sqrt(3) ; you cannot have a square root on the denominator so we rationalize by multiplying by sqrt(3) / sqrt(3) this doesn't change anything because sqrt(3) / sqrt(3) = 1

We then get:
cot(pi/3) = sqrt(3)/3

The cosecant can be figured out much the same way...
we will shorten cosecant to csc
csc(pi/3) = 1/sin(pi/3)
pi/3 rad to degrees again => 60 degrees
1/sin(60) look at a reference triangle (in sources).
sin(60) = opposite / hypotenuse = sqrt(3)/2 / 1 = sqrt(3)/2
csc(60) = 1/sin(60) = 1/sqrt(3) / 2 ; you cant have 1/sqrt(3)/2 so multiply by 2/1 (the reciprocal)
1/sqrt(3) * 2/1
2/sqrt(3) we must rationalize. Multiply by sqrt(3) / sqrt(3).

cot(pi/3) = 2sqrt(3)/3