The period of the business cycle in which the real GDP is increasing is called the?
The trough is not a period, it is simply the lowest point. The opposite of this is a peak, i.e. the highest point. The answer you are looking for is "Expansion" or "Boom". The period in which GDP is falling is called "Recession" or "Contraction".

Also note that the business cycle is typically not defined using real GDP, but rather nominal GDP.