CONTENTS. II. Trigonometrical Ratios of Angles greater than 90°. Relations amongst the Ratios. Change in Sign and Magnitude of the III. Trigonometrical Ratios of the Sums and Differences of Angles, and VI. Logarithms. Their use. Advantage of Logarithms to Base 10. Arrangement of the Tables. Logarithms of the Trigonometrical PLANE TRIGONOMETRY. SECTION I. UNITS OF MEASUREMENT. USE OF SIGNS + AND - MEANING OF THE TERM ANGLE" IN TRIGONOMETRY. NOMETRICAL RATIOS. PRACTICAL APPLICATION. THE TRIGO INSTRU 1. Object of Trigonometry. In Trigonometry we apply Algebraical symbols to establish certain relations between the magnitudes of the sides and angles of plane rectilineal figures. These relations are useful for all the higher branches of Mathematics, and are specially applicable to surveying, and the determining the heights and distances of inaccessible objects. In the present treatise we shall confine ourselves to the simpler relations, and the practical application of them. We must first consider the mode of estimating algebraically the magnitudes of lines, areas, and angles. 2. Measurement of lines. As lines have neither breadth nor thickness we have only to measure their length. To do this we take some standard length, as one foot, one inch, five inches, or any other definite length as our unit of measurement, and the length of any line is then measured and represented by the number, whether whole or fractional, of these units which it contains. Thus if 5 inches is our unit, a line of 20 inches is measured by the number 4, and is called the line 4, that is, the line whose length is 4 times the unit of length. So also the line a would be the line whose length is a times the unit of length. In investigations merely involving algebraical symbols it is indifferent what unit is used, only we must be careful to B. T. 1 remember that throughout the same investigation we use the same unit. Thus if two lines a, b enter into our calculations, we must consider the length of the line a to be a times some unit of length, and of the line b to be b times the same unit. When we apply our general results to numerical examples we must be careful to consider the special unit employed. 3. In the measurement of superficies we take the square of which one side is the unit of length, as the unit of superficies; and then any area is measured by the number of these units it contains. N.B. The area of a rectangle whose sides contain a, b linear units respectively, contains ab superficial units. Also the areas of a parallelogram and a triangle whose bases are b and altitudes a are ab and ab respectively. 4. Measurement of angles. Since the idea of a right angle is simple, and all right angles are equal, it is conveniently taken as a standard by which to measure the magnitudes of other angles. But since from the magnitude of a right angle most of the angles we have to deal with would be represented by fractions or decimals, it is found convenient to divide it. In England we divide the right angle into 90 equal parts called degrees; each degree is divided into 60 minutes; each minute into 60 seconds. Any angle is then measured by the number of degrees, minutes, seconds, and decimal parts of a second it contains; an angle containing 27 degrees, 13 minutes, 24.53 seconds is written thus, 27°, 13', 24".53. In France the decimal system is adopted. A right angle contains 100 grades, a grade 100 minutes, a minute 100 seconds; an angle containing 33 grades, 27 minutes, 45.5 seconds is written thus, 33, 27, 45".5. The advantage of this system is, that any angle can be reduced to the decimal of a grade at once, and vice versa. Thus 33', 27', 45".5=33o.27455, and 27.3567927", 35', 67".9. 5. As an illustration of the use of different units, we will shew how to change our unit from degrees to grades, and vice versa. |