#sec = 1/cos#
Let's first convert to degrees from radians. The conversion for radians to degrees #180/pi#.
#180/pi xx -pi/3#
#= -60^@#
To make this a positive angle, we must subtract 60 from 360, giving us #300^@#. This is a special angle, meaning that it gives us an exact answer. However, before applying our special triangle, we must do this by finding the reference angle. A reference angle is the angle between the terminal side of #theta# to the x axis. It must always satisfy the interval #0^@ <= beta < 90^@#. The closest x axis interception of #300^@# is at #360^@#. Subtracting, we get a reference angle of #60^@#.
We use the #30-60-90, 1, sqrt(3), 2#. Since #60^@# is the reference angle, and 60 is larger than 30, this means the side opposite our reference angle measures #sqrt(3)#. The hypotenuse always is longest; measuring 2. We can now conclude that the adjacent side measures 1.
Applying the definition of cos:
adjacent/hypotenuse = #-1/2 # (cos is negative in quadrant IV)
Substituting into sec.
1/(adjacent/hypotenuse) = hypotenuse/adjacent #= -2#
So, #sec(-pi/3) = -2#
Hopefully this helps!
