Precalculus, Chapter 4, 4.4, Section 4.4, Problem 94a

Since sec is just 1/cos, sec(θ)=2 can also be written as 1/cos(θ)=2.
By multiplying and dividing, you'll get to cos(θ)=1/2.
On the unit circle, always think of the x axis when referring to cosine. At the 1/2 mark on the x axis, draw a vertical line that is parallel to the y axis. Where this vertical line and the circle meet are your 2 answers between 0 and 360 degrees and 0 and 2pi.
In the first quadrant, there is your first solution— 60°, or pi/3 radians.
Your second solution is in the fourth quadrant— 300°, or 5pi/3 radians.
The attached unit circle chart may be of use.