No, you do not need to know the cost basis if you are using the actual expenses or standard mileage method to determine vehicle cost. If you ever want to switch between standard mileage rate and actual expenses you have to you used the standard mileage rate for the first year your vehicle was in service. After that, you can calculate both methods and use whichever gives you a larger deduction. If you use the actual expense method the first year, you must use it for the life of the vehicle.
