From basic trigonometry, we know that;
cosine = Adjacent Side / Hypotenuse
Substituting the known values, we have;
cos(v) = HD / SD
Solving for HD by multiplying both sides of the equation by SD, we get;
HD = SDcos(v)
If the zenith angle (z) is measured rather than the vertical angle, the calculations are nearly identical. The only variation is that the zenith angle is the complimentary angle of the vertical angle, so the sine function must be used. The formula is;
HD = SDsin(z)
Geometronics