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Amy Ferguson Moncure

IB AA · Unit 2 · 5C Applications of Trigonometry

Bearings

Add the north and east parts of a journey, then find the straight trip back to the start.

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Key Ideas Easy Medium Hard

North = d cos(bearing)

East = d sin(bearing)

Bearing is clockwise from north

What to Remember

A bearing

A bearing is measured clockwise from north, from 000° to 360°.

Write it as three digits, such as 045°.

Components

North component = distance × cos(bearing)

East component = distance × sin(bearing)

South and west components are negative.

The trip home

Add the components of each leg.

Distance back = square root of north² + east²

The return bearing points along the opposite of that total displacement.

Easy

North, south, east, or west

Each leg is along a compass direction.

Example
A boat sails 8 km due north, then 6 km due east. Find the distance and bearing of the return trip to the start.

Take north and east as the positive directions.

Leg 1: north 8.00 km, east 0 km

Leg 2: north 0.000000000000000367 km, east 6.00 km

Total north = 8.00 km

Total east = 6.00 km

Distance back = square root of (8.00)² + (6.00)²

Distance back = 10.0 km

The return direction uses the opposite displacement.

Return bearing = 217°

Answer: 10.0 km on a bearing of 217°

1
A drone flies 5 km on a bearing of 000°, then 12 km on a bearing of 090°. Find the distance back to the start and the bearing of that return trip.
Show Answer

Take north and east as the positive directions.

Leg 1: north 5.00 km, east 0 km

Leg 2: north 0.000000000000000735 km, east 12.0 km

Total north = 5.00 km

Total east = 12.0 km

Distance back = square root of (5.00)² + (12.0)²

Distance back = 13.0 km

The return direction uses the opposite displacement.

Return bearing = 247°

Answer: 13.0 km on a bearing of 247°

2
A drone flies 9 km on a bearing of 090°, then 12 km on a bearing of 180°. Find the distance back to the start and the bearing of that return trip.
Show Answer

Take north and east as the positive directions.

Leg 1: north 0.000000000000000551 km, east 9.00 km

Leg 2: north −12.0 km, east 0.00000000000000147 km

Total north = −12.0 km

Total east = 9.00 km

Distance back = square root of (−12.0)² + (9.00)²

Distance back = 15.0 km

The return direction uses the opposite displacement.

Return bearing = 323°

Answer: 15.0 km on a bearing of 323°

3
A drone flies 7 km on a bearing of 000°, then 7 km on a bearing of 270°. Find the distance back to the start and the bearing of that return trip.
Show Answer

Take north and east as the positive directions.

Leg 1: north 7.00 km, east 0 km

Leg 2: north −0.00000000000000129 km, east −7.00 km

Total north = 7.00 km

Total east = −7.00 km

Distance back = square root of (7.00)² + (−7.00)²

Distance back = 9.90 km

The return direction uses the opposite displacement.

Return bearing = 135°

Answer: 9.90 km on a bearing of 135°

4
A drone flies 15 km on a bearing of 180°, then 8 km on a bearing of 090°. Find the distance back to the start and the bearing of that return trip.
Show Answer

Take north and east as the positive directions.

Leg 1: north −15.0 km, east 0.00000000000000184 km

Leg 2: north 0.000000000000000490 km, east 8.00 km

Total north = −15.0 km

Total east = 8.00 km

Distance back = square root of (−15.0)² + (8.00)²

Distance back = 17.0 km

The return direction uses the opposite displacement.

Return bearing = 332°

Answer: 17.0 km on a bearing of 332°

5
A drone flies 10 km on a bearing of 270°, then 10 km on a bearing of 000°. Find the distance back to the start and the bearing of that return trip.
Show Answer

Take north and east as the positive directions.

Leg 1: north −0.00000000000000184 km, east −10.0 km

Leg 2: north 10.0 km, east 0 km

Total north = 10.0 km

Total east = −10.0 km

Distance back = square root of (10.0)² + (−10.0)²

Distance back = 14.1 km

The return direction uses the opposite displacement.

Return bearing = 135°

Answer: 14.1 km on a bearing of 135°

Medium

Two legs

Resolve each bearing, then add.

Example
A drone flies 4.2 km on a bearing of 040°, then 5.6 km on a bearing of 150°. Find the distance back to the start and the bearing of that return trip.

Take north and east as the positive directions.

Leg 1: north 3.22 km, east 2.70 km

Leg 2: north −4.85 km, east 2.80 km

Total north = −1.63 km

Total east = 5.50 km

Distance back = square root of (−1.63)² + (5.50)²

Distance back = 5.74 km

The return direction uses the opposite displacement.

Return bearing = 287°

Answer: 5.74 km on a bearing of 287°

1
A drone flies 3.5 km on a bearing of 025°, then 4.8 km on a bearing of 110°. Find the distance back to the start and the bearing of that return trip.
Show Answer

Take north and east as the positive directions.

Leg 1: north 3.17 km, east 1.48 km

Leg 2: north −1.64 km, east 4.51 km

Total north = 1.53 km

Total east = 5.99 km

Distance back = square root of (1.53)² + (5.99)²

Distance back = 6.18 km

The return direction uses the opposite displacement.

Return bearing = 256°

Answer: 6.18 km on a bearing of 256°

2
A drone flies 6.4 km on a bearing of 070°, then 2.7 km on a bearing of 160°. Find the distance back to the start and the bearing of that return trip.
Show Answer

Take north and east as the positive directions.

Leg 1: north 2.19 km, east 6.01 km

Leg 2: north −2.54 km, east 0.923 km

Total north = −0.348 km

Total east = 6.94 km

Distance back = square root of (−0.348)² + (6.94)²

Distance back = 6.95 km

The return direction uses the opposite displacement.

Return bearing = 273°

Answer: 6.95 km on a bearing of 273°

3
A drone flies 2.8 km on a bearing of 200°, then 3.3 km on a bearing of 290°. Find the distance back to the start and the bearing of that return trip.
Show Answer

Take north and east as the positive directions.

Leg 1: north −2.63 km, east −0.958 km

Leg 2: north 1.13 km, east −3.10 km

Total north = −1.50 km

Total east = −4.06 km

Distance back = square root of (−1.50)² + (−4.06)²

Distance back = 4.33 km

The return direction uses the opposite displacement.

Return bearing = 070°

Answer: 4.33 km on a bearing of 070°

4
A drone flies 5 km on a bearing of 015°, then 5 km on a bearing of 105°. Find the distance back to the start and the bearing of that return trip.
Show Answer

Take north and east as the positive directions.

Leg 1: north 4.83 km, east 1.29 km

Leg 2: north −1.29 km, east 4.83 km

Total north = 3.54 km

Total east = 6.12 km

Distance back = square root of (3.54)² + (6.12)²

Distance back = 7.07 km

The return direction uses the opposite displacement.

Return bearing = 240°

Answer: 7.07 km on a bearing of 240°

5
A drone flies 8.1 km on a bearing of 330°, then 4.4 km on a bearing of 250°. Find the distance back to the start and the bearing of that return trip.
Show Answer

Take north and east as the positive directions.

Leg 1: north 7.01 km, east −4.05 km

Leg 2: north −1.50 km, east −4.13 km

Total north = 5.51 km

Total east = −8.18 km

Distance back = square root of (5.51)² + (−8.18)²

Distance back = 9.87 km

The return direction uses the opposite displacement.

Return bearing = 124°

Answer: 9.87 km on a bearing of 124°

Hard

Three legs

Add every leg before you find the return.

Example
A drone flies 2.4 km on a bearing of 035°, then 3.1 km on a bearing of 140°, then 1.8 km on a bearing of 230°. Find the distance back to the start and the bearing of that return trip.

Take north and east as the positive directions.

Leg 1: north 1.97 km, east 1.38 km

Leg 2: north −2.37 km, east 1.99 km

Leg 3: north −1.16 km, east −1.38 km

Total north = −1.57 km

Total east = 1.99 km

Distance back = square root of (−1.57)² + (1.99)²

Distance back = 2.53 km

The return direction uses the opposite displacement.

Return bearing = 308°

Answer: 2.53 km on a bearing of 308°

1
A drone flies 5 km on a bearing of 040°, then 3.6 km on a bearing of 150°, then 4.2 km on a bearing of 250°. Find the distance back to the start and the bearing of that return trip.
Show Answer

Take north and east as the positive directions.

Leg 1: north 3.83 km, east 3.21 km

Leg 2: north −3.12 km, east 1.80 km

Leg 3: north −1.44 km, east −3.95 km

Total north = −0.724 km

Total east = 1.07 km

Distance back = square root of (−0.724)² + (1.07)²

Distance back = 1.29 km

The return direction uses the opposite displacement.

Return bearing = 304°

Answer: 1.29 km on a bearing of 304°

2
A drone flies 1.5 km on a bearing of 080°, then 2.6 km on a bearing of 170°, then 3.4 km on a bearing of 300°. Find the distance back to the start and the bearing of that return trip.
Show Answer

Take north and east as the positive directions.

Leg 1: north 0.260 km, east 1.48 km

Leg 2: north −2.56 km, east 0.451 km

Leg 3: north 1.70 km, east −2.94 km

Total north = −0.600 km

Total east = −1.02 km

Distance back = square root of (−0.600)² + (−1.02)²

Distance back = 1.18 km

The return direction uses the opposite displacement.

Return bearing = 059°

Answer: 1.18 km on a bearing of 059°

3
A drone flies 6 km on a bearing of 020°, then 4 km on a bearing of 095°. Find the distance back to the start and the bearing of that return trip.
Show Answer

Take north and east as the positive directions.

Leg 1: north 5.64 km, east 2.05 km

Leg 2: north −0.349 km, east 3.98 km

Total north = 5.29 km

Total east = 6.04 km

Distance back = square root of (5.29)² + (6.04)²

Distance back = 8.03 km

The return direction uses the opposite displacement.

Return bearing = 229°

Answer: 8.03 km on a bearing of 229°

4
A drone flies 2.6 km on a bearing of 000°, then 1.9 km on a bearing of 128°. Find the distance back to the start and the bearing of that return trip.
Show Answer

Take north and east as the positive directions.

Leg 1: north 2.60 km, east 0 km

Leg 2: north −1.17 km, east 1.50 km

Total north = 1.43 km

Total east = 1.50 km

Distance back = square root of (1.43)² + (1.50)²

Distance back = 2.07 km

The return direction uses the opposite displacement.

Return bearing = 226°

Answer: 2.07 km on a bearing of 226°

5
A drone flies 2.2 km on a bearing of 125°, then 1.7 km on a bearing of 040°, then 2.9 km on a bearing of 210°. Find the distance back to the start and the bearing of that return trip.
Show Answer

Take north and east as the positive directions.

Leg 1: north −1.26 km, east 1.80 km

Leg 2: north 1.30 km, east 1.09 km

Leg 3: north −2.51 km, east −1.45 km

Total north = −2.47 km

Total east = 1.44 km

Distance back = square root of (−2.47)² + (1.44)²

Distance back = 2.86 km

The return direction uses the opposite displacement.

Return bearing = 330°

Answer: 2.86 km on a bearing of 330°

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