How can I know where to shoot a projectile to make it hit a point?

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I have got two objects in a world. One of them is an archer, and I would like him to shoot the other one in a 3D space.

How do I know what X, Y, and Z velocities to give the arrow? I obviously know about sines and cosines and stuff, but for some reason I seem to be unable to work out how to use them to do this.