It is seen that r is the hypotenuse of a right triangle of which x and y are the legs; hence r2 quadrant the point is located. = x2 + y2, no matter in what EXERCISES 1. Plot the points (4, 5), (2, 7), (0, 4), (5, 5), (7, 0), (−2, 4), (−4, 5), (−6, −2), (0, −7), (−6, 0), (3, −4), (7, −6). 2. Find the radius vector for each of the points (7, 6), (-5, 6), (−4, −6), (9, −8), (9, -10). Plot in each case. Ans. 9.2195; 7.8102; 7.2111; 12.0416; 13.4536. 3. Where are all the points whose abscissas are 2? Whose ordinates are 3? Whose abscissas are -5? Whose radii vectores are 6? 4. The positive direction of the x-axis is taken as the initial side of an angle of 45°. A point is taken on the terminal side with a radius vector equal to 10. Find the ordinate and abscissa of the point. Ans. Each is 7.071. 5. In Exercise 4 what is the ratio of the ordinate to the abscissa? The ratio of the radius vector to the ordinate? Show that you get the same ratios if any other point on the terminal side is taken. 6. The radius vector of a point is 8 and makes an angle of 60° with the positive x-axis. Find the coördinates of the point. Find the coördinates of the points if the radius is the same length and the angles are respectively 120°, 150°, 240°, and 330°. Ans. (4, 4 √3); (-4, 4 √3); (−4 √3, 4); (-4, −4 √3); (4 √3, -4). 7. With the positive x-axis as initial side construct angles of 30°, 150°, 210°, and 330°. Take a point on the terminal side so that the radius vector is 2 a in each case, and find the length of the ordinate and the abscissa of the point. 8. The hour hand of a clock is 4 ft. long. Find the coördinates of its outer end when it is twelve o'clock; when one o'clock; when two; five; eight; ten; half-past four. Use perpendicular and horizontal axes intersecting where the hands are fastened. Ans. (0, 4); (2,2 √3); (2 √3, 2); (2, −2 √3); (−2 √3, −2); (−2 √3, 2); (2 √2, -2√2). CHAPTER II TRIGONOMETRIC FUNCTIONS OF ONE ANGLE 10. Functions of an angle. - Connected with any angle there are six ratios that are of fundamental importance, as upon them is founded the whole subject of trigonometry. They are called trigonometric ratios or trigonometric functions of the angle. One of the first things to be done in trigonometry is to investigate the properties of these ratios, and to establish relations between them. FIG. 12. 11. Trigonometric ratios. Draw an angle in each of the four quadrants as shown in Fig. 12, each angle having its vertex at the origin and its initial side coinciding with the positive part of the x-axis. Choose any point P (x, y) in the terminal side of such angle at the distance r from the origin. Draw MP10X, forming the coördinates OM = x and MP = y, and the radius vector, or distance, OP = r. Then in whatever quadrant @ is found, the functions are defined as follows: sine (written sin 0) A ordinate MP cosine (written cos 0) fore subject to the ordinary rules of algebra, such as addition, subtraction, multiplication, and division. 12. To each and every angle there corresponds but one value of each trigonometric ratio. Draw any angle as in Fig. 13. Choose points P1, P2, P3, etc., on the terminal side OP. Draw From the geom MIP1, MP2, MP3, etc., perpendicular to OX. and similarly for the other trigonometric ratios. Hence the six ratios remain unchanged as long as the value of the angle is unchanged. Definitions. When one quantity so depends on another that for every value of the first there are one or more values of the second, the second is said to be a function of the first. Since to every value of the angle there corresponds a value for each of the trigonometric ratios, the ratios are called trigonometric functions. They are also called natural trigonometric functions in order to distinguish them from logarithmic trigonometric functions. A table of natural trigonometric functions for angles from 0° to 90° for each minute is given on pages 108 to 130 of Tables.* An explanation of the table is given on page 27 of Tables. 13. Signs of the trigonometric functions. The sine of an angle has been defined as the ratio of the ordinate to the distance of any point in the terminal side of the angle. Since the distance r is always positive (Art. 9), sin 0 will have the same algebraic sign as the ordinate of the point. Therefore sin is positive when the angle is in the first or second quadrant, and negative when the angle is in the third or fourth quadrant. In a similar manner the algebraic signs of the remaining functions of are determined. The student should verify the following table: 14. Exponents of trigonometric functions. Y When the trigo nometric functions are to be raised to powers they are written sin2 0, cos3 0, tan1 0, etc., instead of (sin 0)2, (cos 0)3, (tan 0)4, etc., except when the exponent is -1. Then the function is enclosed in parentheses. Thus, (sin )-1= (see Art. 26). 1 sin 0 15. Calculation of trigonometric functions by measurement. 23/16 25° X M FIG. 14. Functions of 25°. By means of the protractor draw angle XOP 25°, Fig. 14. Choose P in the terminal side, say, 2 = *The reference is to Logarithmic and Trigonometric Tables by the Authors. Exercise. Construct several acute angles at random, with their vertices at the origin and their initial sides on the positive part of the x-axis. Choose a point in the terminal side of each angle, draw and measure its coördinates, and calculate the trigonometric Y 0° FIG. 15. functions for each angle. Measure each angle with the protractor and determine its trigonometric functions from the Table IV. Compare these functions P(a) with those computed. X 0°. The initial and terminal sides of 0° are both on OX. Choose the point P on OX as in Fig. 15, at the distance of a from O. Then the coördinates of P are (a, 0). By definition then we have: 0. *cot 0° 1. = 0. *csc 0° = a У a = tan 0° * By the expression as a limit. For example, 0.0000001 That is, as x gets nearer and nearer to zero gets larger and larger, and can be made to become larger than any number N. The value of is then said to become infinite as x approaches zero. The |