180 degrees= pi radians

sin 11pi/6=sin 11*180/6=sin 11 30 degrees=sin 330 degrees=-sin 30 degrees=-1/2

-sin 30 is in the fourth quadrant

draw a right triangle with 30 degrees at the vertex and one leg part of the positive x-axis and the other leg perpendicular to the positive x-axis

sin=opposite/hypotenuse

sin(-30)=-1/2 so the perpendicular side is -1 and the hypotenuse is 2

therefore the other leg is, from the Pythagorean Theorem a^2+b^2=c^2, a^2+1=4 and a^2=3 giving us

a=sqrt(3)

cos(-30)=adjacent/hypotenuse=-sqrt(3)/2

sec=1/cos

sec=1/[-sqrt(3)/2]=-2/sqrt(3)

we got the answer from the value of sin 11pi/6

however we could have done the following:

sec 7pi/6=sec 7*30=sec 210=-sec 30=-2sqrt(3)/3=-2/sqrt(3) [multiply both terms by sqrt(3)]

-sec 30 is in the third quadrant going clockwise