Elevation, depression and bearings

Angles of elevation and depression, true bearings, and solving practical problems with trigonometry and Pythagoras.

Elevation and depression

The angle of elevation is measured up from the horizontal; the angle of depression is measured down from the horizontal. They are equal (alternate angles between parallel horizontal lines).

Example 1●●

From $50$ m away, the angle of elevation to the top of a tower is $32^\circ$. Find its height.

$h = 50\tan 32^\circ \approx 31.24$ m

Example 2●●●

From an $80$ m cliff, the angle of depression to a boat is $25^\circ$. How far is the boat from the base of the cliff?

$\tan 25^\circ = \dfrac{80}{d}$

$d = \dfrac{80}{\tan 25^\circ} \approx 171.56$ m

True bearings

A true bearing is an angle measured clockwise from north, written with three digits: east is $090^\circ$, south is $180^\circ$, south-west is $225^\circ$.

Flashcards

Angle of elevation

Up from the horizontal

True bearing of west

$270^\circ$

Back bearing

Add or subtract $180^\circ$.

Practice

From $50$ m away, the angle of elevation to a tower top is $32^\circ$. Height (2 d.p.)?
From an $80$ m cliff, depression to a boat $25^\circ$. Distance from the base (2 d.p.)?
True bearing of south-west?
Back bearing of $060^\circ$?
True bearing of east (as a number)?
A ramp rises at $15^\circ$ over a horizontal distance of $100$ m. Height gained (2 d.p.)?
The angle of depression from A to B equals:
A hiker walks $5$ km on a bearing of $090^\circ$, then $3$ km on $180^\circ$. Distance from the start (2 d.p.)?