The values of apparent angles of dip in two places measured in two mutually perpendicular planes are `30^(@)` and `45^(@)`. Determine the true angle of dip at the place

Explanation

Let `phi` be the true angle of dip and `phi_(1) , phi_(2)` be the apparent angles of dip in two mutually perpendicular planes such that `phi_(1) = 30^(@)` and `phi_(2) = 45^(@) phi = ?`
Consider the figure shown below

If one vertical plane is at angle `theta` with magnetic meridian , the other vertical plane makes angle `90^(@) - theta` with the magnetic meridian .
The apparent angle of dip is related to true angle of dip as
`tan phi' = (tan phi)/(cos theta)`
So, we can write
`tan phi_(1) = (tan phi)/(cos theta) " " ... (i)`
and `tan phi_(2) = (tan phi)/(sin theta) " " ... (ii)`
As we know that
`sin^(2) theta + cos^(2) theta = 1`
So , we get
`(tan^(2) phi)/(tan^(2) phi_(1)) + (tan^(2) phi)/(tan^(2) phi_(2)) = 1`
`implies cot^(2) phi_(1) + cot^(2) phi_(2) = cot^(2) phi`
Plugging in the values of `phi_(1)` and `phi_(2)` , we get
`cot^(2) 30^(@) + cot^(2) 45^(@) = cot^(2) phi`
`implies cot^(2) phi = 3 + 1`
= 4
`cot phi = 2`
`phi = cot^(-1) (2)`
= `tan^(-1)(1)/(2)`.