'How to convert slope in inches per foot to degrees' and 'find the distance between two lines that are at an angle of 114 degrees at any given point from the apex' are different questions.
How to convert slope in inches per foot to degrees:
From the Wikipedia page on sine (sin) ( http://en.wikipedia.org/wiki/Sine ), we have this:
Inches are 'a' (opposite) and feet are (b) adjacent.
You'll have to convert to the same units, something like inches per inch. But since the number of feet in 'inches per foot' is 1, b is 12 inches instead of 1 foot.
But sin(alpha) is a/h, so we need h.
It's a right triangle, so h = sqrt(12^2 + a^2)
sin(alpha) = a / sqrt(144+a^2)
alpha = arcsin ( a / sqrt(144+a^2) )