In mathematics, a bearing is defined as an angle measured clockwise from north. Bearings are usually written as a three-figure bearing. For example, the angle 50° from north is written as 050°.

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A ship leaves point A on a bearing of 45° and travels for 13 km. Another ship leaves point A on a bearing of 155° and travels for 20 km. How far apart are the two ships?

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To find a bearing using trigonometry, create a right-angled triangle. If distances are given in any of the compass directions, label these sides as the adjacent or opposite sides of the triangle. If a distance is given in a particular bearing, label this distance as the hypotenuse of the triangle.

In a quadrant bearing, the angle is always measured as the smallest angle from either north or south. This angle can be either clockwise or counter-clockwise depending on which direction is the nearest. The angle of a true bearing is always measured from north and the angle is always given in a clockwise direction.

Therefore a quadrant bearing of S30°W would convert to a true bearing of 210°. This is because the angle measured clockwise from north would include the 180° to face south, then a further 30°. In total, 180° + 30° = 210°.

In this case, the bearing of B from A and the given angle of 80° both form interior angles. Therefore they both add to 180°.

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In this example, the cosine rule is used to find a missing side length and then the sine rule is used to find a missing angle. This angle is then used to find the bearing.

Since the two known sides of the triangle are the opposite and adjacent, we can use the tan function to find the angle shown.

We can see that alternate angles are equal. Therefore the angle between the line AB and the horizontal is 20° as shown below.

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Whilst the relative bearing is 010°, the true bearing of this new direction is 060° because the ship is already on a bearing of 50° and 50° + 10° = 60°.

The relative bearing is the angle between the direction of travel and some other direction. This angle is measured clockwise from the current direction.

A bearing is always defined as an angle clockwise from north. Bearings are important for navigation in 2-dimensions such as when using maps. Bearings are useful because the angle is always relative to north and therefore they provide a consistent measurement of direction.

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For any vector of the form ai+bj, the angle of the vector is found using tan-1(b/a). The bearing of the vector is then found as the angle clockwise from north. For example, the vector (3i-4j) makes an angle of tan-1(4/3) = 53° below east. The bearing is found as 90°+53°=143°.

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For example, a ship is currently travelling on a bearing of 050°. A relative bearing of 010° is made from the ship’s current direction.

In Step 2, an interior angle of the triangle is found. Firstly, we use the fact that interior angles add to 180° to find the 160° angle marked in blue. Secondly, angles around a point add to 360°, so subtracting 160° and 170° from 360°, we see that the internal angle of the triangle must be 30°.

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The true bearing is the angle measured clockwise from north. The true bearing is often referred to simply as the bearing.

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Since the distances given are in compass directions of east and north, a right-angled triangle can be drawn in which the 10 km east is the adjacent side and the 6 km north is the opposite side.

A quadrant bearing is the angle made between a north or south direction and an east or west direction. North or south is written first, then the angle, then east or west. For example, a quadrant bearing of S30°W means to start facing south then make a 30° angle towards the west.

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To convert a quadrant bearing to a true bearing, first draw a diagram of the quadrant bearing. The true bearing is the angle measured clockwise from north to this bearing. Use the fact that from north, east is 90°, south is 180° and west is 270°. Then add or subtract the quadrant bearing angle from these values depending on if the quadrant bearing is measured clockwise or counter-clockwise respectively.

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Alternate angles are equal and are recognised by the ‘z’ shape made between the two angles. These angle facts can be used to calculate bearings.

To calculate a bearing, find the angle clockwise from north. Start by drawing a vertical line representing north at the first location. Draw a line connecting the start location to the end location. Measure the clockwise angle between the north line and this line.

The bearing of the final position from the starting position must be given as clockwise from north. The bearing is indicated with the angle shown below.

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Since the distances given are in the compass directions of west and south, a right-angled triangle can be formed with an adjacent of 8 km and an opposite of 3 km.

If bearings are made between three locations and a non-right-angled triangle is formed between them, then the sine rule or cosine rule, c2=a2+b2-2ab.cos(C) may be used to solve the problem. The distances travelled make up the side lengths a, b and c and the internal angles of the triangle are A, B and C.

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Start by drawing a diagram. 300° clockwise from north is shown below. It is past 270° but less than 360° and so, it lies between west and north.

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Interior angles add to 180°. If given the bearing from B to A, the bearing from A to B can be found using interior angles. Subtract the bearing of B to A from 180° to find the missing interior angle, then use the fact that angles in a full turn add to 360° to find the bearing of A to B.