Hello, Fibonacci!!

1. The pilot of an aircraft flying due South, at an altitude of 750m,

sees an airfield directly ahead, angle of depression 17°.

At the same time the pilot notices a town due east, angle of depression 10°.

How far is town from airfield? Once you "see" the situation, make separate diagrams.

Code:

P * - - - - - - - - - - - - S
| * 17°
| *
750 | *
| *
| 17° *
Q * - - - - - - - - - - - * A
750/tan17°

The pilot is at $\displaystyle P$ at an altitude of 750 m: .$\displaystyle PQ = 750$

The airfield is at $\displaystyle A.\;\;\angle SPA \:=\:\angle PAQ \:=\:17^o$

$\displaystyle \tan17^o \:=\:\frac{750}{AQ}\quad\Rightarrow\quad AQ \:=\:\frac{750}{\tan17^o}$

Code:

P * - - - - - - - - - - - - E
| * 10°
| *
750 | *
| *
| 10° *
Q * - - - - - - - - - - - * T
750/tan10°

The pilot is at P: .$\displaystyle PQ = 750$

The town is at $\displaystyle T.\;\;\;\angle EPT = \angle PTQ = 10^o$

$\displaystyle \tan10^o \:=\:\frac{750}{QT}\quad\Rightarrow\quad QT \:=\:\frac{750}{\tan10^o}$

Looking straight down at the ground, we have this diagram: Code:

750/tan10°
P * - - - - - - - - * T
| *
| *
750/tan17° | *
| *
| *
A *

I assume you can find the distance $\displaystyle AT$ now.