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I have heard that it's possible to dampen it with glue, like is possible on a graphics card, but that would definitely void the warranty. I suppose I will just have to get used to it for now.
In video cards coil whine is often an effect of very high FPS or screen refresh rates with a light rendering load. It's very common especially when browsing web sites in browsers that use GPU rendering so make sure it's not your GPU.
GPU drivers usually allow you to change the display adapter refresh rate: right click on desktop/display settings/advanced display settings/display adapter properties for Display 1/Monitor tab/select Screen refresh rate from the pick list. If lowering it to something like 60hz then it's more likely your GPU.
But if it is the CPU VRM: putting a thermal pad on the VRM...and pay particular attention to the the coils, those large square black compononents...might help. But first press a pencil eraser on the coils to see if you can affect the sound when you hear the whining before going to the trouble and expense. If it changes, you've found the culprit. Also touch (but a not as firmly) the electrolytic cap's too. If nothing changes it then neither thermal pads nor glueing them will be likely to help.
You might also try making minor changes to VCore or LLC settings. Something like increasing/decreasing VCore by one notch using offset adjustment or using one notch higher or lower of LLC. The idea is to change the how the VRM is delivering voltage to the CPU in the useage scenario where it whines.
Some motherboards have more extensive settings like VRM duty cycle and PRF (pulse recurring frequency) in BIOS, which would be best. Those things have very wide margins for adjustment so very slight changes shouldn't affect performance or stability adversely. Except when overclocked and/or undervolted to the point it's on a razor's edge for stability, of course.