APPLICATIONS TO THE CELESTIAL AND TERRESTRIAL SPHERES Astronomical Problems ANSWERS TO EXERCISES . PLANE TRIGONOMETRY FIG. I CHAPTER I THE TRIGONOMETRIC FUNCTIONS ANGLES 1. In Trigonometry the size of an angle is measured by the amount one side of the angle has revolved from the position of the other side to reach its final position. Thus, if the hand of a clock makes one-fourth of a revolution, the angle through which it turns is one right angle; if it makes one-half a revolution, the angle is two right angles; if one revolution, the angle is four right angles; if one and one-half revolutions, the angle is six right angles, etc. The amount the side OB has rotated from OA to reach its final position may or may not be equal to the inclination of the lines. In Fig. I it is equal to this inclination; in Fig. 4 it is not. Two angles may have the same sides and yet be different. In Fig. 2 and Fig. 4 the positions of the sides of the angles are the same; yet in Fig. 2 the angle is two right angles, in Fig. 4 it is six right angles. The addition of any number of complete revolutions to an angle does not change the position of its sides. Question.-Through how many right angles does the hour-hand of a clock revolve in 6 hours? the minute-hand? Question. If the fly-wheel of an engine makes 100 revolutions per minute, through how many right angles does it revolve in I second? Terminal line Terminal line Initial line RIGHT ANGLES - Def. The first side of the angle—that is, the side from which the revolution is measured-is the initial line; the second side is the terminal line. Terminal line Def. If the direction of the revolution is opposite to that of the hands of a clock, the angle is positive; if the same as that of the hands of a clock, the angle is negative. Initial line POSITIVE ANGLE Initial line 5 RIGHT ANGLES Terminal line Initial line NEGATIVE ANGLE The angles we have employed as illustrations-those described by the hands of a clock--are all negative angles. 2. Angles are usually measured in degrees, minutes, and seconds. A degree is one-ninetieth of a right angle, a minute is one-sixtieth of a degree, a second is one-sixtieth of a O /// The symbols indicating degrees, minutes, and seconds are thus, twenty-six degrees, forty-three minutes, and ten seconds is written 26° 43′ 10′′. II 3. The plane about the vertex of an angle is divided into four quadrants, as shown in the figure; the first quadrant begins at the initial line. III II I THE FOUR QUADRANTS III IV Terminal Line ANGLE IN 3D QUADRANT II III ANGLE IN 4TH QUADRANT IV An angle is said to be in a certain quadrant if its terminal line is in that quadrant. EXERCISES 4. (1.) Express 24 right angles in degrees, minutes, and seconds. In what quadrant is the angle? (2.) What angle less than 360° has the same initial and terminal lines as an angle of 745°? (3.) What positive angles less than 720° have the same sides as an angle of 73°? (4.) In what quadrant is an angle of — 890°? |