OF TRIGONOMETRY, PLAIN AND SPHERICAL. BY EDWARD OLNEY, PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF MICHIGAN. NEW YORK: 667 BROADWAY. STODDARD'S JUVENILE MENTAL ARITHMETIC SHORT AND FULL COURSE FOR GRADED SCHOOLS. STODDARD'S PICTORIAL PRIMARY ARITHMETIC STODDARD'S COMPLETE ARITHMETIC 30 75 1 25 The Combination School Arithmetic being Mental and Written Arithmetic in one book, will alone serve for District Schools.. For Academies a full high course is obtained by the Complete Arithmetic and Intellectual Arithmetic. STODDARD'S HIGHER MATHEMATICS. A COMPLETE SCHOOL ALGEBRA in one vol., 390 pages, $1.50. Designed for A GEOMETRY AND TRIGONOMETRY in one vol. By Prof. EDWARD OLNEY. In press. A GENERAL GEOMETRY AND CALCULUS in one vol. In press. The other books of Stoddard's Series will be published as rapidly as possible. Entered according to Act of Congress, in the year 1870, BY SHELDON & COMPANY, In the Office of the Librarian of Congress, at Washington. PART IV. TRIGONOMETRY. CHAPTER I PLANE TRIGONOMETRY. SECTION I DEFINITIONS AND FUNDAMENTAL RELATIONS BETWEEN THE TRIGONOMETRICAL FUNCTIONS OF AN ANGLE (OR ARC). 1. Trigonometry is a part of Geometry which has for its subject-matter, Angles. It is chiefly occupied in presenting a scheme for measuring and comparing angles, by means of certain auxiliary lines called Trigonometrical Functions, in investigating the relations between these functions, and in the solution of triangles by means of the relations between their sides and the trigonometrical functions of their angles. 2. A Function is a quantity, or a mathematical expression, conceived as depending upon some other quantity or quantities for its value. = ILL'S.-A man's wages for a given time is a function of the amount received. per day; or, in general, his wages is a function of both the time of service and: the amount received per day. Again, in the expressions y 2ax2, y 2:3 2bx +5, y = 2 log ax, y = a*, y is a function of x; since, the numbers 2, 5, a and b being considered fixed or constant, the value of y depends upon the value we assign to x. For a like reason such expressions as √a2 — x2, and 3ax2 — 2√x, may be spoken of as functions of x. Once more, the area of a triangle is a function of its base and altitude. 1 3. Angles as Functions of Arcs.-We have learned in Geometry (Part II.), that angles and arcs may be treated as functions of each other; and that, if the angles be taken at the centre of the same or equal circles, the arcs intercepted have the same ratio as the angles themselves, and hence may be taken as their measures or representatives. For trigonometrical purposes, an angle is considered as measured by an arc struck with a radius 1, from the angular point as a centre. 4. A Degree being the part of the circumference of a circle, becomes the measure of of a right angle; and, for convenience, it is customary to speak of such an angle as an angle of one degree, four times as large an angle as an angle of four degrees, etc., applying the term directly to the angle. A small circle written at the right and a little above a number indicates degrees (°). 5. A Minute is part of a degree. Minutes are designated by an accent ('). A Second is part of a minute. Seconds are indicated by a double accent ("). Smaller divisions of angles (or arcs) are most conveniently represented as decimals of a second, though the designations thirds, fourths, etc., are sometimes met with, and signify further subdivisions into 60ths. 5° 12' 16" 13"" is read, "5 degrees, 12 minutes, 16 seconds, and 13 thirds." ILL'S.-In Fig. 1 AOB is an angle of 35°, because the measuring arc ab 7. The Complement of an angle or arc is what remains after subtracting the angle or arc from 90°. 8. The Supplement of an angle or arc is what remains after subtracting the angle or arc from 180°. ILL'S.-In Fig. 1, the angle BOD is the complement of AOB, and the arc bd is the complement of arc ab. The complement of 35° is 90° — 35°— 55°. The supplement of 35° is 180° 35° = 145°. - 9. A Quadrant is often represented by, since is the semicircumference when the radius is unity. When this notation is used, the Unit Arc becomes 180° = T 570.29578 nearly, or 57° 17′ 44′′.8 +, which is an arc equal in length to the radius. 10. For trigonometrical purposes, an angle is conceived as generated by the revolution of a line about the angular point, and hence may have any value whatever, not only from 0° to 180°, but from 0° to 360°, and even to any number of degrees greater than 360°, as 1280°, etc. An angle of 45° is generated by of a revolution, 90° by of a revolution, 180° by a revolution, 270° by 4 of a revolution, 360° by one revolution, 450° by 14 revolutions, 1280° by 3 revolutions, etc., etc. 11. In accordance with the conception of an angle as generated by a revolving line, the measuring arc is considered as originating at the first position of the revolving line (i. e., with one side of the angle), and terminating in the line after it has generated the angle under consideration (i. e., with the other side of the angle). The first extremity is called the Origin of the arc, and the other the Termination. ILL'S.-In Fig. 1, let the angle AOB be considered as generated by a line starting from the position OA, and revolving around the point O, from right to left,* till it reaches the position OB. Oa being taken as unity, the arc ab is the measuring arc of the angle AOB; a is its origin, and b its termination. 12. In the generation of angles by means of a revolving line, the normal motion is considered to be from right to left, and the quadrants are numbered 1st, 2d, 3d, and 4th, in the order in which they are generated. 13. The Trigonometrical Functions are eight in number; viz., sine, cosine, tangent, cotangent, secant, cosecant, versedsine, and coversed-sine. These lines are functions of angles, or, what amounts to the same thing, of arcs considered as measures of angles, and are the characteristic quantities of trigonometry. 14. The Sine of an angle (or arc) is a perpendicular let fall from the termination of the measuring arc upon the diameter passing through the origin of the arc. Thus in Fig. 2, bd is in each case the sine of the angle AOB, or of the arc axb. * The pupil will understand that, if he imagin es himself standing at the centre of motion, as the moving body or point passes before him, the distinctions "from right to left," and from left" to right, are easily made. |