Right-angled trigonometry, elevation and depression

A quick review of SOH CAH TOA, then angles of elevation and depression, eye height, and problems that use two right-angled triangles.

Before you can handle triangles without a right angle, the right-angled ones need to be automatic. This page is a fast review, then the two kinds of angle that HSC questions love to describe in words: elevation and depression. Revise: the Year 10 trigonometry notes cover SOH CAH TOA from scratch if you need them.

SOH CAH TOA in one minute

The three ratios

$$\sin\theta = \frac{O}{H} \qquad \cos\theta = \frac{A}{H} \qquad \tan\theta = \frac{O}{A}$$

Label the sides from the angle you are using: the hypotenuse is always opposite the right angle, the opposite side faces your angle, and the adjacent side touches it. Then pick the ratio that links the side you know with the side you want. If you want an angle, use the inverse key: $\theta = \tan^{-1}\left(\dfrac{O}{A}\right)$.

Example 1●

A kite flies on a $50$ m string. The kite is $32$ m above the ground. Find the angle the string makes with the ground, to the nearest degree.

$O = 32$, $H = 50$, so use sine.

The string is the hypotenuse. The height is opposite the angle at the ground.

$\theta = \sin^{-1}\left(\dfrac{32}{50}\right) \approx 39.8^\circ$

We want an angle, so use the inverse.

The string makes an angle of about $40^\circ$ with the ground.

Round only at the end.

Angles of elevation and depression

Both angles are measured from the horizontal, never from the vertical.

The angle of elevation is how far you tilt your eyes up from horizontal to see something above you.
The angle of depression is how far you tilt your eyes down from horizontal to see something below you.

Here is the useful fact. If you stand on a cliff and look down at a boat, and someone in the boat looks up at you, the two angles are equal. The horizontal line at the top of the cliff and the water are parallel, so the angle of depression and the angle of elevation are alternate angles. That lets you move the angle into the right-angled triangle at the bottom, where it is easier to use.

InteractiveMeasuring with angles

Try this

In Angle of elevation mode, set the distance to $35$ m and the angle to $38^\circ$. Check Example 2. Now double the distance and keep the tree the same height: does the angle halve? (It doesn't. Tangent is not a straight line.)

Switch to Angle of depression. Set the cliff to $12$ m and the angle to $9^\circ$ to check Example 3. Watch the blue angle at the boat: it always matches the yellow one.

Example 2●

Ana stands $35$ m from the base of a Moreton Bay fig in the Royal Botanic Garden. The angle of elevation of the top of the tree is $38^\circ$, and her eyes are $1.6$ m above the ground. How tall is the tree?

Let $h$ be the height above her eyes. $\tan 38^\circ = \dfrac{h}{35}$

Draw the horizontal from her eyes, not from the ground. The right-angled triangle sits above eye level.

$h = 35\tan 38^\circ \approx 27.34$ m

Multiply both sides by 35.

Tree height $\approx 27.34 + 1.6 \approx 28.9$ m

The triangle stopped at eye level, so add the eye height back on.

Example 3●●

A lifeguard tower at Bondi has its lookout $12$ m above the water. The angle of depression from the lookout to a swimmer is $9^\circ$. How far is the swimmer from the base of the tower?

In the triangle at the water, the angle at the swimmer is $9^\circ$.

Alternate angles: the angle of elevation from the swimmer up to the lookout is also 9°.

$\tan 9^\circ = \dfrac{12}{d}$

Height is opposite that angle; the distance is adjacent to it.

$d = \dfrac{12}{\tan 9^\circ} \approx 75.8$ m

The unknown is on the bottom, so swap it with tan 9°.

Problems with two right-angled triangles

Many harder questions have two triangles that share a side. The method is always the same: find the shared side from the first triangle, then use it in the second. Keep the shared value unrounded in your calculator.

Example 4●●●

From a lookout $80$ m above the sea, the angles of depression to two boats, both due east of the lookout, are $32^\circ$ and $21^\circ$. How far apart are the boats?

Near boat: $d_1 = \dfrac{80}{\tan 32^\circ} \approx 128.03$ m

The nearer boat has the bigger angle of depression: you look further down to see it.

Far boat: $d_2 = \dfrac{80}{\tan 21^\circ} \approx 208.41$ m

Same height, smaller angle.

Distance apart $= 208.41 - 128.03 \approx 80.4$ m

Both distances are measured from the base of the lookout along the same line, so subtract.

Your turn

A drone hovers above a sports field. From a point $90$ m across the field from the spot directly below the drone, the angle of elevation of the drone is $24^\circ$. How high is the drone? Ignore eye height.

Show the working

The height is opposite $24^\circ$ and the $90$ m is adjacent. $h = 90\tan 24^\circ \approx 40.1$ m.

Flashcards

Angle of elevation

Measured up from the horizontal to the line of sight

Angle of depression

Measured down from the horizontal to the line of sight. It equals the angle of elevation from the other end (alternate angles).

Unknown on the bottom: $ an heta = dfrac{h}{d}$

$d = \dfrac{h}{\tan\theta}$

Practice

An angle of elevation is measured:
From $50$ m away, the angle of elevation of the top of a tower is $28^\circ$. Find the height of the tower above eye level, to one decimal place.
A boat is $300$ m from the base of a $45$ m cliff. Find the angle of depression from the top of the cliff to the boat, to one decimal place.
Why does the angle of depression from $A$ to $B$ equal the angle of elevation from $B$ to $A$?
A $6$ m ladder leans against a wall, making an angle of $72^\circ$ with the ground. How far up the wall does it reach? Answer to two decimal places.
A kite string is $40$ m long and makes an angle of elevation of $55^\circ$. How high is the kite, to one decimal place? Ignore the height of the person.
A lighthouse is $h$ m tall. The angle of depression from the top to a ship is $\theta$. Which expression gives the distance $d$ from the base to the ship?
Exam style: Jordan stands $60$ m from the base of an office tower in Parramatta. The angle of elevation of the top is $41^\circ$ and his eyes are $1.7$ m above the ground. Find the height of the tower to one decimal place.
Exam style: From the top of a $65$ m cliff, the angles of depression of two buoys in a straight line out to sea are $35^\circ$ and $18^\circ$. How far apart are the buoys, to one decimal place?