The whine when it turns off is from the coolant heat storage water pump (aka thermos water pump). The warranty you bought from the used car dealer will likely be invalid even if you have a breakdown--third party warranties generally aren't worth the paper they're printed on.
There is a TSB on this: T-SB-0087-08
According to the TSB: This repair is covered under the Toyota Powertrain Warranty. This warranty is in effect for 60 months or 60,000 miles, whichever occurs first, from the vehicle?s in-service date.
Call the Toyota hotline for more information. 1-800-331-4331 (You will need the VIN number).
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