Precise rotation (steering direction to a given angle) on obstacle avoidance robot
The only way you can know how far you have turned, with any form of accuracy and, more importantly, repeatability, is by employing some form of feedback.

In other words, to work out how far you have turned you need, first, to know how far you have turned.

The simplest method is probably to use some form of monitoring on the wheels themselves. That could be an optical encoder to give a nice high resolution "number of degrees the wheel has turned", which can be coupled with the circumference of the wheel to calculate the distance travelled (which can then be used with some simple trigonometry to work out the angle you have turned through) or a more coarse "magnet on the wheel + hall effect sensor" to give you the number of revolutions the wheel has gone through (not very accurate and only really any use at higher speeds when there will be lots of revolutions. Maybe this method would be better on the motor before the gearbox that drives the wheel?)

However, do you really care that much anyway? Are you over thinking things a little too much?

Surely a simpler method would be to first turn left an arbitrary amount (somewhere around 60-degrees-ish, you're not really worried) then sweep from left to right (a distance of somewhere around 120 degrees in total) looking for the furthest distance. Once you have completed your sweep you then just reverse your turn direction until you read the same (or within a certain tolerance) distance at which point you will be facing your longest space. 

All that can be done just with spinning the wheels either for a pre-defined period, or until a specific reading has been obtained. No need to worry about precise angles or anything fancy like that.