Solve a navigation problem with bearings

Bearing of B from A

\(\displaystyle 052^{\circ}\)

Find the marked angle. The diagram is not to scale

Compare bearings of 25 and 65 degrees. Then try other positions. Watch the two distances while the journey length stays fixed

As the bearing increases towards east, how do the two distances change?

Match the reference line to the triangle

First locate the acute angle between the journey and the reference line. A perpendicular distance is positive; a signed component also records direction

1. 1Locate the angle A bearing starts at north. An angle measured from an east-west line may be its complement. In other quadrants, locate the nearby cardinal ray before subtracting
2. 2Find a distance from a line When the acute angle is measured from the line, the perpendicular distance is opposite it and the journey is the hypotenuse \(\displaystyle \sin\theta=\frac{d}{L}\)
3. 3Find the other component The component along that same line is adjacent to the acute angle \(\displaystyle \cos\theta=\frac{a}{L}\)
4. 4Keep direction separate from magnitude For signed northward and eastward components, south is negative northward and west is negative eastward. Calculate a positive length first, then apply its direction

Bearing of B from A

\(\displaystyle 037^{\circ}\)

Journey length AB

\(\displaystyle 18.6\ \mathrm{km}\)

Calculate the shortest distance from B to the east-west line, to one decimal place. The diagram is not to scale

The shortest distance meets the line at a right angle. Find the angle between the journey and east

Bearing of B from A

\(\displaystyle 331^{\circ}\)

Journey length AB

\(\displaystyle 14.2\ \mathrm{km}\)

Calculate the signed eastward displacement of B from A, to one decimal place. Use a negative value for west. The diagram is not to scale

Find the acute angle from north, calculate the westward length, then apply the requested sign

When the journey length is not given

A speed and a duration can supply the hypotenuse. Match their units first. Then use the same bearing and triangle method

1. 1Match time units Convert minutes to hours when speed is in kilometres per hour \(\displaystyle t_{\mathrm{hours}}=\frac{t_{\mathrm{minutes}}}{60}\)
2. 2Find the journey length Multiply the speed by the duration in matching units \(\displaystyle L=vt\)
3. 3Round once Keep the full journey length and calculator values until you round the requested final distance

Bearing of B from A

\(\displaystyle 139^{\circ}\)

Speed

\(\displaystyle 4.2\ \mathrm{km/h}\)

Duration

\(\displaystyle 180\ \mathrm{min}\)

A walker travels from A to B at constant speed. Calculate the shortest distance from B to the east-west line, to one decimal place. The diagram is not to scale

First find the journey length in kilometres. The acute angle from east is the bearing minus east's bearing

Bearing of B from A

\(\displaystyle 041^{\circ}\)

Journey length AB

\(\displaystyle 22.4\ \mathrm{km}\)

A student calculates BP. Which statement identifies the first error? The diagram is not to scale