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Forgive my lack of knowledge of some robotic related terms, but the way my two wheel drive robot steers is simply by having each wheel move at different speeds; for left, for example, the left wheel moves while the right one stays stationary for a given delay of time and vice versa. but I want to make my obstacle detection a little more sophisticated with the following logic:

As the robot moves, scan the environment using an ultrasonic sensor(60 degrees to left and 60 degrees to the right) and using a simple max finding algorithm find the optimal angle i.e. angle at which the ultrasonic reading was highest. and then I want to find a way, using the same system of vehicle rotation if possible, turn to that specific angle proceed to move forward and scan simultaneously.

what I aim to achieve in the end is smoother obstacle avoidance as compared to the most common stop when close to obstacle -> turn left or right -> move and loop

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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.

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You could try an electronic compass, such as this one. Use it to compare your current (turn) angle to the desired angle, and keep turning until the desired angle is reached.

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