The value of apparent angles of dip at two places measured in two mutually perpendicular planes are deg30 and deg45. The true angle of dip at the place is

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)`.