A TREATISE ON TRIGONOMETRY: PART I. PLANE TRIGONOMETRY. CHAPTER I. MEASUREMENT OF ANGLES. 1. Trigonometry is that branch of mathematics which treats (1) of the solution of plane and spherical triangles, and (2) of the general relations of angles and certain functions of them called the trigonometric functions. Plane Trigonometry comprises the solution of plane triangles and investigations of plane angles and their functions. Trigonometry was originally the science which treated only of the sides and angles of plane and spherical triangles; but it has been recently extended so as to include the analytic treatment of all theorems involving the consideration of angular magnitudes. 2. The Measure of a Quantity. - All measurements of lines, angles, etc., are made in terms of some fixed standard or unit, and the measure of a quantity is the number of times the quantity contains the unit. It is evident that the same quantity will be represented by different numbers when different units are adopted. For example, the distance of a mile will be represented by the number 1 when a mile is the unit of length, by the number 1760 when a yard is the unit of length, by the number 5280 when a foot is the unit of length, and so on. In like man 1 mers the Runikiei: expressing the magnitude of an angle will depend on the unit af angle. EXAMPLES. 1. What is the measure of 21 miles when a yard is the unit? 24 miles = { x 1760 yards =4400 yards = 4400 x 1 yard. .. the measure is 4400 when a yard is the unit. = 2. What is the measure of a mile when a chain of 66 feet is the unit? Ans. 80. 3. What is the measure of 2 acres when a square whose side is 22 yards is the unit ? Ans. 20. 4. The measure of a certain field is 44 and the unit is 1100 square yards; express the area of the field in acres. Ans. 10 acres. 5. If 7 inches be taken as the unit of length, by what number will 15 feet 2 inches be represented ? Ans. 26. 6. If 192 square inches be represented by the number 12, what is the unit of linear measurement ? Ans. 4 inches. 3. Angles. — An angle is the opening between two straight lines drawn from the same point. The point is called the vertex of the angle, and the straight lines are called the sides of the angle. An angle may be generated by revolving a line from coincidence with another line about a fixed point. The initial and final positions of the line are the В sides of the angle; the amount of revolution measures the magnitude of the angle; and the angle may be traced out by any number of revolutions of the line. А Thus, to form the angle AOB, OB may be supposed to have revolved from OA to OB; and it is obvious that OB may go on revolving until it comes into the same position OB as many times as we please; the angle AOB, having the same bounding lines OA and OB, may therefore be greater than 2, 4, 8, or any number of right angles. The line QA from which OB moves is called the initial line, and OB in its final position, the terminal line. The revolving line OB is called the generatrix. The point O is called the origin, vertex, or pole. 4. Positive and Negative Angles. — We supposed in Art. 3 that OB revolved in the direction opposite to that of the hands of a watch. But angles may, of course, be described by a line revolving in the same direction as the hands of a watch, and it is often necessary to distinguish between the two directions in which angles may be measured from the same fixed line. This is conveniently effected by adopting the convention that angles measured in one direction shall be considered positive, and angles measured in the opposite direction, negative. In all branches of mathematics angles described by the revolution of a straight line in the direction opposite to that in which the hands of a watch move are usually considered positive, and all angles described by the revolution of a straight line in the same direction as the hands of a watch move are considered negative. Thus, the revolving line OB starts from the initial line OA. When it revolves in the B direction contrary to that of the hands of a watch, and comes into the position OB, it traces out the positive angle AOB (marked + a); A and when it revolves in the same direction as the hands of a watch, it traces the negative angle AOB (marked — b). The revolving line is always considered negative. 5. The Measure of Angles. — An angle is measured by the arc of a circle whose centre is at the vertex of the ta P, BA' 66 angle and whose ends are on the sides of the angle (Geom., Art. 236). Let the line OP of fixed length generate an angle by revolving in the positive direction B round a fixed point 0 from an initial Pi P. position OA. Since OP is of constant length, the point P will trace out the circumference ABA'B' whose centre A is 0. The two perpendicular diameters AA' and BB' of this circle will inclose the four right angles AOB, В" BOA', A'OB', and B'OA. The circumference is divided at the points A, B, A', B' into four quadrants, of which AB is called the first quadrant. second quadrant. A'B'66 third quadrant. B'A " " fourth quadrant. In the figure, the angle AOP, between the initial line OA and the revolving line OP, is less than a right angle, and is said to be an angle in the first quadrant. AOP, is greater than one and less than two right angles, and is said to be an angle in the second quadrant. AOP, is greater than two . and less than three right angles, and is said to be an angle in the third quadrant. AOP, is greater than three and less than four right angles, and is said to be an angle in the fourth quadrant. When the revolving line returns to the initial position 0A, the angle AOA is an angle of four right angles. By supposing OP to continue revolving, the angle described will become greater than an angle of four right angles. Thus, when OP coincides with the lines OB, OA', OB', OA, in the second revolution, the angles described, measured from the beginning of the first revolution, are angles of five right angles, six right angles, seven right angles, eight right 1 |