C. Trigonometric Functions · Trigonometry of Acute Triangles

Angles of Elevation and Depression, and Bearings

C1.1About 70 minutes

Learning goals

I can identify angles of elevation and depression and explain why they are equal.
I can read and write directions as true bearings (such as 065°) and as compass (quadrant) bearings (such as N 65° E).
I can solve surveying and navigation problems with one right triangle.
I can draw a clear, labelled sketch from a word problem.

Words to know

angle of elevation
The angle between the horizontal and the line of sight when you look UP at something.

angle of depression
The angle between the horizontal and the line of sight when you look DOWN at something.

line of sight
The straight line from an observer's eye to the object being viewed.

true bearing
A direction measured clockwise from north, written with three digits: from 000° up to (but not including) 360°. East is 090°.

compass (quadrant) bearing
A direction written as N or S, then an angle, then E or W. N 65° E means start facing north and turn 65° toward east.

Looking up, looking down, finding the way

Surveyors, pilots, search-and-rescue crews and ferry captains all describe directions with angles. Two kinds come up all the time:

Vertical angles: how far up or down you look from the horizontal.
Horizontal directions: which way you are heading, measured from north.

Both turn real situations into right triangles you already know how to solve.

Elevation and depression

An angle of elevation is measured up from the horizontal. An angle of depression is measured down from the horizontal. Both are always measured from a horizontal line, never from a vertical wall or a cliff face.

If a person on the ground looks up at a person on a cliff, and the person on the cliff looks back down, the two angles are equal: the two horizontal lines are parallel and the line of sight cuts across them, making alternate angles.

Both angles are measured from a horizontal line. Because the horizontals are parallel, the angle of elevation from below equals the angle of depression from above.

Worked example

Example 1: A kite on a windy day

Sam holds a kite string 1.2 m above the ground. The string is 60 m long and makes an angle of elevation of 48°. Assuming the string is straight, how high is the kite above the ground?

Sketch a right triangle from Sam's hand: hypotenuse 60 m (the string), angle 48° at the hand.
Height above the hand is opposite: 60 sin 48° ≈ 44.59 m.
Add the height of the hand: 44.59 + 1.2 ≈ 45.8 m.

The kite is about 45.8 m above the ground.

Worked example

Example 2: Angle of depression from a cliff

From the top of a 48 m cliff above a lake, the angle of depression to a canoe is 18°. How far is the canoe from the base of the cliff?

The angle of depression is measured from the horizontal at the top. By alternate angles, the angle of elevation from the canoe up to the cliff top is also 18°.
In the right triangle at the canoe: the cliff (48 m) is opposite 18° and the distance d is adjacent.
tan 18° = 48/d, so d = 48 ÷ tan 18° ≈ 147.7 m.

The canoe is about 148 m from the base of the cliff.

Example 2. The 18° angle of depression at the top equals the 18° angle of elevation at the canoe, so the right triangle at the base can be solved with tangent. (The cliff height is drawn to scale with the distance.)

Bearings

There are two common ways to give a direction on a map:

A true bearing is measured clockwise from north, using three digits. North is 000°, east is 090°, south is 180°, west is 270°.
A compass (quadrant) bearing starts at N or S and turns toward E or W. N 65° E is the same direction as 065°. S 40° W means face south, then turn 40° toward west; that is 180° + 40° = 220°.

In navigation problems, draw a small north arrow at every point where the direction changes. The angle you need is usually between a north–south line and the path.

Left: true bearing 065° is 65° clockwise from north, the same as N 65° E. Right: S 40° W is 40° from south toward west, the same as true bearing 220°.

Worked example

Example 3: How far north and east?

A float plane leaves a dock and flies 220 km on a bearing of 040°. How far north and how far east of the dock is it?

Draw north at the dock. The path makes a 40° angle with north. Drop a perpendicular from the plane to the north line to make a right triangle.
The northward distance is adjacent to 40°: 220 cos 40° ≈ 168.5 km.
The eastward distance is opposite 40°: 220 sin 40° ≈ 141.4 km.

The plane is about 169 km north and 141 km east of the dock.

Worked example

Example 4: Distance and bearing back

A sailboat travels 12 km due east and then 5 km due south. How far is it from where it started, and on what true bearing would it need to sail to go straight there from the start?

The two legs meet at a right angle. Distance = √(12² + 5²) = √169 = 13 km.
At the start, the angle south of east is tan⁻¹(5/12) ≈ 22.6°.
East is 090°. Turning a further 22.6° clockwise (toward south) gives 090° + 22.6° ≈ 113°.

The boat is 13 km from the start, on a true bearing of about 113° (compass bearing S 67° E).

Just for fun

A funny-but-true fact: the north end of a compass needle points to magnetic north, which is not the same place as the geographic North Pole, and it slowly moves over the years. That is why navigators adjust their compass readings to get true bearings.

Hands-on activity

Measure a tall object with a homemade clinometer

You need: a protractor, a drinking straw, tape, string, a small weight (a metal washer or a key), a tape measure and a partner. Never look toward the sun through the straw.

Tape the straw along the straight edge of the protractor. Tie the string through the protractor's centre hole and hang the weight from it.
Look through the straw at the top of a tall object (a house, a basketball net, a tree). Your partner reads where the string crosses the scale. The angle of elevation is the difference between that reading and 90° (the string reads 90° when the straw is level).
Measure your distance from the base of the object and the height of your eye above the ground.
Height = distance × tan(elevation) + eye height.

What to look for: if you can, check the object's height another way (for a basketball net, the rim is usually 3.05 m). Repeat from a second distance; the two heights should agree within about 10%.

Your turn

From 35 m away from the base of a building, the angle of elevation to the roof is 52°. How tall is the building? (Measure from ground level.)

From an apartment window 22 m up, the angle of depression to a parked car is 34°. How far is the car from the base of the building?

Write the true bearing for: (a) N 30° W (b) S 15° E (c) due west.

Write the compass bearing for: (a) 125° (b) 250°.

A ship sails 40 km on a bearing of 120°. How far south and how far east of its starting point is it?

A 30 m tall cell tower is anchored by a cable from the top to a point 14 m from its base. Find the angle of elevation of the cable.

A hiker walks 3.0 km north and then 4.5 km west. Find her distance and true bearing from the start.

More practice: warm-up

Is an angle of depression measured from the horizontal or from the vertical?

What is the true bearing of due south?

Write 090° as a direction word.

The angle of elevation from A to B is 25°. What is the angle of depression from B to A?

Convert N 45° E to a true bearing.

More practice: on your own

A drone hovers 80 m above a field. A person standing on the field sees it at an angle of elevation of 27° (ignore eye height). How far is the person from the point directly below the drone?

A lighthouse keeper is 36 m above the water and sees a boat at an angle of depression of 9°. How far is the boat from the base of the lighthouse? Then: a minute later the angle is 14°. Is the boat getting closer or farther?

A plane flies 350 km on a bearing of 300°. How far north and how far west of its start is it?

Jordan says the angle of depression from the top of a 20 m tower to a point 20 m from its base is 45°, and the angle of elevation from that point to the top is also 45°. Is he right?

The angle of elevation to the top of a 12 m tall maple tree is 31°, measured with eyes 1.5 m above the ground. How far is the observer from the tree?

Challenge

A boat leaves port and sails 18 km on a bearing of 050°. How far, and on what true bearing, must it sail to return directly to port?

From the top of a 60 m building, the angle of depression to the top of a shorter building 45 m away (measured horizontally) is 20°. How tall is the shorter building?

A ship is 15 km due east of a lighthouse. A second ship is due north of the same lighthouse, on a bearing of 330° from the first ship. How far apart are the ships, and how far is the second ship from the lighthouse?

Check your understanding

Quiz

An angle of elevation is measured:

Why is the angle of depression from B to A equal to the angle of elevation from A to B?

What is the true bearing for S 40° W?

From the top of a 48 m cliff, the angle of depression to a canoe is 18°. Which gives the horizontal distance d?

A plane flies 220 km on a bearing of 040°. How far east has it gone?

True bearing 270° is the direction:

For parents and teachers

Big idea: This lesson applies C1.1 to the real-world contexts named in the curriculum: surveying (heights and distances) and navigation (bearings).

How to help: Insist on a sketch with a clearly drawn horizontal line at every observer and a north arrow at every turning point. Most mistakes come from measuring an angle from the wrong line.

Common difficulties:

Measuring an angle of depression from the vertical cliff face instead of the horizontal.
Forgetting eye height when an observer is standing.
Confusing true bearings (clockwise from north) with angles measured from the x-axis in math class (counterclockwise from east).
For the clinometer activity: remind your teen never to sight the sun.