130. Middle Latitude Sailing. Since a ship rarely sails for any length of time due east or due west, the difference in longitude cannot ordinarily be found as in parallel sailing (§§ 128, 129). Therefore, in plane sailing the departure between two places is measured generally on that parallel of latitude which lies midway between the parallels of the two places. This is called the method of middle latitude sailing. Hence, in middle latitude sailing, This assumption produces no great error, except in very high latitudes or excessive runs. Exercise 66. Middle Latitude Sailing 1. A ship leaves latitude 31° 14' N., longitude 42° 19' W., and sails E.N.E. 32 mi. Find the position reached. 2. Leaving latitude 49° 57' N., longitude 15° 16' W., a ship sails between S. and W. till the departure is 38 mi. and the latitude is 49° 38' N. Find the course, distance, and longitude reached. 3. Leaving latitude 42° 30' N., longitude 58° 51' W., a ship sails S.E. by S. 48 mi. Find the position reached. 4. Leaving latitude 49° 57' N., longitude 30° W., a ship sails S. 39° W. and reaches latitude 49° 44' N. Find the distance and the longitude reached. 5. Leaving latitude 37° N., longitude 32° 16' W., a ship sails between N. and W. 45 mi. and reaches latitude 37° 10' N. Find the course and the longitude reached. 6. A ship sails from latitude 40° 28' N., longitude 74° W., on an E.S.E. course, 62 mi. Find the latitude and longitude reached. 7. A ship sails from latitude 42° 20' N., longitude 71° 4' W., on a N.N.E. course, 30 mi. Find the latitude and longitude reached. 131. Traverse Sailing. In case a ship sails from one point to another on two or more different courses, the departure and difference of longitude are found by reckon ing each course separately and combining the results. For example, two such courses are shown in the figure. This is called the method of traverse sailing. No new principles are involved in traverse sailing, as will be seen in solving Ex. 1, given below. Exercise 67. Traverse Sailing 1. Leaving latitude 37° 16' S., longitude 18° 42' W., a ship sails N.E. 104 mi., then N.N.W. 60 mi., then W. by S. 216 mi. Find the position reached, and its bearing and distance from the point left. For the first course we have difference of latitude 73.5 N., departure 73.5 E.; for the second course, difference of latitude 55.4 N., departure 23 W.; for the third course, difference of latitude 42.1 S., departure 211.8 W. On the whole, then, the ship has made 128.9 mi. of north latitude and 42.1 mi. of south latitude. The place reached is therefore on a parallel of latitude 86.8 mi. to the north of the parallel left; that is, in latitude 35° 49.2' S. In the same way the departure is found to be 161.3 mi. W., and the middle latitude is 36° 32.6'. With these data we find the difference of longitude to be 201', or 3° 21′ W. Hence the longitude reached is 22° 3′ W. With the difference of latitude 86.8 mi. and the departure 161.3 mi., we find the course to be N. 61° 43′ W. and the distance 183.2 mi. The ship has reached the same point that it would have reached if it had sailed directly on a course N. 61° 43′ W. for a distance of 183.2 mi. 2. A ship leaves Cape Cod (42° 2' N., 70° 3' W.) and sails S.E. by S. 114 mi., then N. by E. 94 mi., then W.N.W. 42 mi. Find its position and the total distance. 3. A ship leaves Cape of Good Hope (34° 22' S., 18° 30' E.) and sails N.W. 126 mi., then N. by E. 84 mi., then W.S. W. 217 mi. Find its position and the total distance. 4. A ship in latitude 40° N. and longitude 67° 4' W. sails N.W. 60 mi., then N. by W. 52 mi., then W.S.W. 83 mi. Find its position. 5. A ship sailed S.S. W. 48 mi., then S.W. by S. 36 mi., and then N.E. 24 mi. Find the difference in latitude and the departure. 6. A ship sailed S. E. 18 mi., S.W. S. 37 mi., and then S.S.W W. 56 mi. Find the difference in latitude and the departure. CHAPTER X GRAPHS OF FUNCTIONS 132. Circular Measure. Besides the methods of measuring angles which have been discussed already and are generally used in practical work, there is another method that is frequently employed in the theoretical treatment of the subject. This takes for the unit the angle subtended by an arc which is equal in length to the radius, and is known as circular measure. 133. Radian. An angle subtended by an arc equal in length to the radius of the circle is called a radian. The term "radian" is applied to both the angle and arc. In the annexed figure we may think of a radius bent around the arc AB so as to coincide with it. Then ZAOB is a radian. 134. Relation of the Radian to Degree Measure. The number of radians in 360° is equal to the B A number of times the length of the radius is contained in the length of the circumference. It is proved in geometry that this number is 2π for all circles, π being equal to 3.1416, nearly. Therefore the radian is the same angle in all circles. The circumference of a circle is 27 times the radius. 135. Number of Radians in an Angle. From the definition of radian we see that the number of radians in an angle is equal to the length of the subtending arc divided by the length of the radius. Thus, if an arc is 6 in. long and the radius of the circle is 4 in., the number of radians in the angle subtended by the arc is 6 in. ÷ 4 in., or 11⁄2. 136. Reduction of Radians and Degrees. From the values found in § 134 the following methods of reduction are evident: To reduce radians to degrees, multiply 57° 17′ 45′′, or 57 29578°, by the number of radians. To reduce degrees to radians, multiply 0.017453 by the number of degrees. These rules need not be learned, since we do not often have to make these reductions. It is essential, however, to know clearly the significance of radian measure, since we shall often use it hereafter. In solving the following problems the rules may be consulted as necessary. In particular the student should learn the following: The word radians is usually understood without being written. Thus sin 2π means the sine of 2π radians, or sin 360°; and tan means the tangent of radians, or 45°. Also, sin 2 means the sine of 2 radians, or sin 114.59156°. State the quadrant in which the following angles lie: Express the following in degrees and also in radians: 25. g of four right angles. 26. § of four right angles. 27. of two right angles. 28. of one right angle. 29. What decimal part of a radian is 1°? 1'? 30. How many minutes in a radian? How many seconds? 31. Express in radians the angle of an equilateral triangl 32. Over what part of a radian does the minute h move in 15 min. ? 137. Functions of Small Angles. Let AOP be any acute angle, and let x be its circular measure. Describe a circle of unit radius about O as center and take ZAOP' =—ZAOP. Draw the tangents to the circle at P and P', meeting OA in 7. Then we see that If, now, the angle x is constantly diminished, cos x approaches the value 1. Accordingly, the limit of sin æ as a approaches 0, is 1. x Hence when x denotes the circular measure of an angle near 0° we may use x instead of sin x and instead of tan x. For example, required to find the sine and cosine of 1'. If x is the circular measure of 1', x = 2 π 360 × 60 3.14159 + = 0.00029088+, the next figure in a being 8. Now sinx>0 but <x; hence sin 1'lies between 0 and 0.000290889. Again, cos 1'= √1- sin'1'>√1-(0.0003) > 0.9999999. .. sin 1' > 0.000290888 × 0.9999999 0.000290888 · 0.0000000000290888 > 0.000290887. Hence sin 1' lies between 0.000290887 and 0.000290889; that is, to eight places of decimals, sin 1'= 0.00029088 +, the next figure being 7 or 8. |