50. Trigonometrical Problems sometimes require a knowledge of the Points of the Mariner's Compass, which we shall now explain.

In the above figure, it will be seen that 32 points are taken

at equal distances on the circumference of a circle, so that the arc between any two consecutive points subtends at the centre of the circle an angle equal to 360°, that is to 11°.

32

The points North, South, East, West are called the Cardinal Points, and with reference to them the other points receive their names. The student will have no difficulty in learning these if he will carefully notice the arrangement in any one of the principal quadrants.

51. Sometimes a slightly different notation is used; thus N. 11° E. means a direction 111° east of north, and is therefore the same as N. by E. Again S.W. by S. is 3 points from south and may be expressed by S. 334° W., or since it is 5 points from west it can also be expressed by W. 561° S. In each of these cases it will be seen that the angular measurement is made from the direction which is first mentioned.

52. The angle between the directions of any two points is obtained by multiplying 111° by the number of intervals between the points. Thus between S. by W. and W.S.W. there are 5 intervals and the angle is 561°; between N.E. by E. and S.E. there are 7 intervals and the angle is 783°.

53. If B lies in a certain direction with respect to A, it is said to bear in that direction from A; thus Birmingham bears N.W. of London, and from Birmingham the bearing of London is S.E.

Example 1. From a lighthouse L two ships A and B are observed in directions S. W. and 15° East of South respectively. At the same time B is observed from A in a S.E. direction. If LA is 4 miles find the distance between the ships.

Example 2. At 9 A. M. a ship which is sailing in a direction E. 40° S. at the rate of 8 miles an hour observes a fort in a direction 50° North of East. At 11 A. M. the fort is observed to bear N. 20° W.: find the distance of the fort from the ship at each observation.

Let A and C be the first and second positions of the ship; B the fort.

Through A draw lines towards the cardinal points of the compass. From the observations made

LEAC=40°, LEAB=50°, so that ▲ BAC=90°.

Through C draw CN' towards the North; then ▲ BCN'=20°, for the bearing of the fort from C is N. 20° W.

.. LACB= LACN' - LBCN'= 50° - 20° 30°.

Thus the distances are 9.237 and 18-475 miles nearly.

EXAMPLES. VI. b.

1. A person walking due E. observes two objects both in the N.E. direction. After walking 800 yards one of the objects is due N. of him, and the other lies N.W.: how far was he from the objects at first?

2. Sailing due E. I observe two ships lying at anchor due S.; after sailing 3 miles the ships bear 60° and 30° S. of W.; how far are they now distant from me?

3. Two vessels leave harbour at noon in directions W. 28° S. and E. 62° S. at the rates 10 and 101 miles per hour respectively. Find their distance apart at 2 p.m.

4. A lighthouse facing N. sends out a fan-shaped beam extending from N.E. to N.W. A steamer sailing due W. first sees the light when 5 miles away from the lighthouse and continues to see it for 30/2 minutes. What is the speed of the steamer?

5. A ship sailing due S. observes two lighthouses in a line exactly W. After sailing 10 miles they are respectively N.W. and W.N.W.; find their distances from the position of the ship at the first observation.

6. Two vessels sail from port in directions N. 35° W. and S. 55° W. at the rates of 8 and 8/3 miles per hour respectively. Find their distance apart at the end of an hour, and the bearing of the second vessel as observed from the first.

7. A vessel sailing S.S.W. is observed at noon to be E.S.E. from a lighthouse 4 miles away. At 1 p.m. the vessel is due S. of the lighthouse: find the rate at which the vessel is sailing. Given tan 67°=2·414.

8._ A, B, C are three places such that from A the bearing of C is N. 10° W., and the bearing of B is N. 50° E.; from B the bearing of C is N. 40° W. If the distance between B and C is 10 miles, find the distances of B and C from A.

9. A ship steaming due E. sights at noon a lighthouse bearing N.E., 15 miles distant; at 1.30 p.m. the lighthouse bears N.W. How many knots per day is the ship making? Given 60 knots=69 miles.

10. At 10 o'clock forenoon a coaster is observed from a lighthouse to bear 9 miles away to N.E. and to be holding a south-easterly course; at 1 p.m. the bearing of the coaster is 15° S. of E. Find the rate of the coaster's sailing and its distance from the lighthouse at the time of the second observation.

11. The distance between two lighthouses, A and B, is 12 miles and the line joining them bears E. 15° N. At midnight a vessel which is sailing S. 15° E. at the rate of 10 miles per hour is N.E. of A and N.W. of B: find to the nearest minute when the vessel crosses the line joining the lighthouses.

12. From A to B, two stations of a railway, the line runs W.S.W. At A a person observes that two spires, whose distance apart is 1.5 miles, are in the same line which bears N.N.W. At B their bearings are N. 7° E. and N. 37° E. Find the rate of a train which runs from A to B in 2 minutes.

CHAPTER VII.

RADIAN OR CIRCULAR MEASURE.

54. We shall now return to the system of measuring angles which was briefly referred to in Art. 6. In this system angles are not measured in terms of a submultiple of the right angle, as in the sexagesimal and centesimal methods, but a certain angle known as a radian is taken as the standard unit, in terms of which all other angles are measured.

55. DEFINITION. A radian is the angle subtended at the centre of any circle by an arc equal in length to the radius of the circle.

B

In the above figure, ABC is a circle, and O its centre. If on the circumference we measure an arc AB equal to the radius and join OA, OB, the angle AOB is a radian.

56. In any system of measurement it is essential that the unit should be always the same. In order to shew that a radian, constructed according to the above definition, is of constant magnitude, we must first establish an important property of the circle.

H. K. E. T.

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