For most woodworking tasks around the house, such as wall framing, fencing or building a garage work table, the great majority of connections will be square joints. In these circumstances, you'll probably never need to set your circular saw greater than 45 degrees to make an acute angle. However, if you're planning on building something a little more elaborate, such as a child's outdoor playhouse, there may be a few times where this will be a necessary skill to have in your arsenal. Below I will describe the procedure to make these more difficult angles.
Suppose we want to make a 60 degree bevel cut on the end of a 1' 2x6 board. You may think it would be alright to start off with a piece identical to the one shown above, but that would be an incorrect assumption. The one shown above is unfortunately too short. Instead, you'll need another inch to work with in addition to your original measurement to make the correct cut.
Because our circular saw can't make a 60 degree cut, we'll instead need to make two cuts that'll equal 60 degrees. First, we'll need to do a little first grade math.
1. Subtract the desired angle from 90 and cut that in the opposite direction.
In our case, that angle would be 30 degrees. This will be the angle we calibrate our circular saw too. When this angle is set, cut the board along the original measurement with the saw blade angled in the opposite direction of the future 60 cut. When finished, it should look like the below illustration. Now you can see why we couldn't just use a 1' board.
2. Make a 90 degree cut along the edge of the bevel
Now that we have out first beveled cut, we can set our circular saw back to zero to make a 2nd, 90 degree cut. Make sure to secure the board firmly before proceeding. Using both hands, run the saw right along the edge of the 30 degree bevel. In other examples, if the angle is steep enough, the circular saw's blade depth may not be enough to reach all the way though and will require a hand saw or reciprocating saw to finish the cut. When the 2nd cut is complete, you'll be left with a 60 degree angle in your board and a little right triangle
This method will work with any angle over 45°. Some circular saws can cut up to 50 degrees and some miter saws can even get to 60°. If you're going to be making a lot of these types of acute angles, or just lots of cuts in general, a good miter saw may be worth the investment.
All of this is assuming, of course, that you're comfortable with working with a circular saw and holding it on strange angles. The traditional way to cut these types of angles is with a good ol' hand saw. Simply mark your angle with a speed square, mark the plumb cut and saw away.