Determine new GPS position using roll, pitch, yaw and length

I'm using a Reach RS+ device to capture GPS position data as well as IMU data (roll, pitch and yaw); see the "Point Collection" image on the manufacturer's website.
I'm trying to determine the GPS coordinates of the bottom point (the empty end of the rod that the receiver is fixed on).
To be able to make calculations in meters I'm converting longitude (X) and latitude (Y) to UTM while keeping altitude (Z) unaltered.
When the rod is upright, X and Y stay the same while

Z1 = Z - ROD_LENGTH  

However when the rod is tilted all coordinates are affected and I need a way to calculate the rod's end position.
I have looked at rotation matrices, triangle equations, my own sin and cos formulas based on experimental observations but I don't have a background in 3D geometry and I'm not sure which route to take (for example I don't know how to use the rod length with a rotation matrix).
Basically I need the following formulas:

X1 = X + ROD_LENGTH * func_X(roll, pitch, yaw)
Y1 = Y + ROD_LENGTH * func_Y(roll, pitch, yaw)
Z1 = Z + ROD_LENGTH * func_Z(roll, pitch, yaw)

Roll, pitch and yaw have values between -180° and 180°.

from Determine new GPS position using roll, pitch, yaw and length