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!
