At an angle of 30$$^\circ$$ to the magnetic meridian, the apparent dip is 45$$^\circ$$. Find the true dip:

The relation between apparent dip $$\delta'$$, true dip $$\delta$$, and the angle $$\phi$$ from the magnetic meridian is $$\tan\delta' = \frac{\tan\delta}{\cos\phi}$$.

Here $$\phi = 30^\circ$$ and $$\delta' = 45^\circ$$, so $$\tan 45^\circ = \frac{\tan\delta}{\cos 30^\circ}$$.

This gives $$1 = \frac{\tan\delta}{\sqrt{3}/2}$$, so $$\tan\delta = \frac{\sqrt{3}}{2}$$.

Therefore the true dip is $$\delta = \tan^{-1}\!\left(\dfrac{\sqrt{3}}{2}\right)$$.