Hello, iiharthero!

A slight variation of Aidan's excellent approach . . .

A man walking due north observes a church spire with bearing 342°.

After walking 1500m, he observes it with bearing 337°.

How far is the church from the road?

I'll use Aidan's diagram . . .

Code:

h
C * - - - - - - - - - * D
* * |
* * | x
* * 23° |
* * |
* * B
* |
* 18° | 1500
* |
* |
* A

The church is at .The man walks along the road

He makes his first observation at

He walks 1500 m to

. . And

The distance from the church to the road is:

. . Let

In right triangle .[1]

In right triangle .[2]

Equate [2] and [1]: .

Multiply by

. . and we have: .

Factor: .

Therefore: .