Elevation & Depression — Mathslive.ie

Section 1 of 2

Angle of Elevation

Looking up

The angle of elevation is the angle you look up from the horizontal to see something.

Worked example — height of a mast

From a point $15\,\text{m}$ from the base of a mast, the angle of elevation to the top is $21^\circ$. Find the height $h$.

Have adj ($15$), want opp ($h$) $\Rightarrow$ $\tan$

$\tan 21 = \dfrac{h}{15} \Rightarrow h = 15\tan 21$

$h = 5.75 = 5.8\,\text{m}$

$h = 5.8\,\text{m}$

Worked example — a ladder

A ladder $8\,\text{m}$ long rests against a wall with its base $3\,\text{m}$ from the wall. Find the angle of elevation the ladder makes with the ground.

Have adj ($3$) and hyp ($8$) $\Rightarrow$ $\cos$

$\cos A = \dfrac{3}{8} \Rightarrow A = 68^\circ$

$A = 68^\circ$

Section 2 of 2

Angle of Depression

Looking down

The angle of depression is the angle you look down from the horizontal. It equals the angle of elevation looking back up (alternate angles).

Worked example — from the top of a building

A building is $41\,\text{m}$ tall. From the top, the angle of depression to a point on the ground is $38^\circ$. Find the distance $x$ of the point from the base.

The angle up from the point equals $38^\circ$; have opp ($41$), want adj ($x$) $\Rightarrow$ $\tan$

$\tan 38 = \dfrac{41}{x} \Rightarrow x = \dfrac{41}{\tan 38}$

$x = 52.5\,\text{m}$

You try

From the top of a cliff $28\,\text{m}$ high, the angle of depression to a boat is $34^\circ$. Find the distance of the boat from the base of the cliff.

$\tan 34 = \dfrac{28}{x}$.

$x = \dfrac{28}{\tan 34} = 41.5\,\text{m}$

$x = 41.5\,\text{m}$

That’s Class 4.

Elevation (up) and depression (down). Class 5: labelled triangles and Pythagoras together.