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TRIGONOMETRIC FUNCTIONS OF ACUTE ANGLES
1. The Nature of Arithmetic. In arithmetic we study computation, the working with numbe 3. We may have a formula expressed in algebraic symbols, such as a =bh, but the actual computation involved in applying such a formula to a particular case is part of arithmetic.
Arithmetic enters into all subsequent branches of mathematics. It plays such an important part in trigonometry that it becomes necessary to introduce another method of computation, the method which makes use of logarithms.
2. The Nature of Algebra. In algebra we generalize arithmetic. Thus, instead of saying that the area of a rectangle with base 4 in. and height 2 in. is 4 x 2 sq. in., we express a general law by saying that a = bh. Algebra, therefore, is a generalized arithmetic, and the equation is the chief object of attention.
Algebra also enters into all subsequent branches of mathematics, and its relation to trigonometry will be found to be very close.
3. The Nature of Geometry. In geometry we study the forms and relations of figures, proving many properties and effecting numerous constructions concerning them.
Geometry, like algebra and arithmetic, enters into the work in trigonometry. Indeed, trigonometry may almost be said to unite arithmetic, algebra, and geometry in one subject.
4. The Nature of Trigonometry. We are now about to begin another branch of mathematics, one not chiefly relating to numbers although it uses numbers, not primarily devoted to equations although using equations, and not concerned principally with the study of geometric forms although freely drawing upon the facts of geometry.
Trigonometry is concerned chiefly with the relation of certain lines in a triangle (trigon, "a triangle," + metrein, " to measure") and forms the basis of the mensuration used in surveying, engineering, mechanics, geodesy, and astronomy.
5. How Angles are Measured. For ordinary purposes angles can be measured with a protractor to a degree of accuracy of about 30'.
The student will find it advantageous to use the convenient protractor furnished with this book and shown in the illustration below.
For work out of doors it is customary to use a transit, an instrument by means of which angles can be measured to minutes. By turning the top of the transit to the right or left, horizontal angles can be measured on the horizontal plate. By turning the telescope up or down, vertical angles can be measured on the vertical circle seen in the illustration.
For astronomical purposes, where great care is necessary in measuring angles, large circles are used.
The degree of accuracy required in measuring an angle depends upon the nature of the problem. We shall now assume that we can measure angles in degrees, minutes, and seconds, or in degrees and decimal parts of a degree. Thus 15° 30' is the same as 15.5°, and 15° 30' 36" is the same as 152° + of 19, or 15.51°.
The ancient Greek astronomers had no good symbols for fractions. The best system they could devise for close approximations was the so-called sexagesimal one, in which there appear only the nunerators of fractions whose denominators are powers of 60. This system seems to have been first suggested by the Babylonians, but to have been developed by the Greeks. It is much inferior to the decimal system that was perfected about 1600, but the world still continues to use it for the measure of angles and time. The decimal division of the angle is, however, gaining ground, and in due time will probably replace the more cumbersome one with which we are familiar.
In this book we shall use both the ancient and modern systems, but with the chief attention to the former, since this is still the more common.
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6. Functions of an Angle. In the annexed figure, if the line AR moves about the point A in the sense indicated by the arrow, from the position AX as an initial position, it generates the angle A. If from the points B, B', B", ...,
on AR, we let fall the perpendiculars BC, B'C', В"C", ..., on AX, we form a series of similar triangles ACB, AC'B', AC"B", and so on. The corresponding sides of these triangles are proportional. That is, BC B'C' B"C"
each of which has a series of other ratios equal to it.
AB AB AB'' For example,
BC B'C' B"C" That is, these ratios remain unchanged so long as the angle remains unchanged, but they change as the angle changes.
Each of the above ratios is therefore a function of the angle A.
As already learned in algebra and geometry, a magnitude which depends upon another magnitude for its value is called a function of the latter magnitude. Thus a circle is a function of the radius, the area of a square is a function of the side, the surface of a sphere is a function of the diameter, and the volume of a pyramid is a function of the base and altitude.
We indicate a function of x by such symbols as f(x), F), f'(x), and $(x), and we read these "f of x, f-major of x, f-prime of x, and phi of x” respectively.
For example, if we are repeatedly using some long expression like ** + 3x – 2x2 + 7x – 4, we may speak of it briefly as f (x). If we are using some function of angle A, we may designate this as f(A). If we wish to speak of some other function of A, we may write it f'(A), F(A), or $(A).
In trigonometry we shall make much use of various functions of an angle, but we shall give to them special names and symbols. On this account the ordinary function symbols of algebra, mentioned above, will not be used frequently in trigonometry, but they will be used often enough to make it necessary that the student should understand their significance.
7. The Six Functions. Since with a given angle A we may take any one of the triangles described in $ 6, we shall consider the triangle ACB, lettered as here shown.
It has long been the custom to letter in this way the hypotenuse, sides, and angles of the first triangle considered in trigonometry, C being the right angle, and the hypotenuse and sides bearing the small letters corresponding to the opposite capitals. By the sides of the triangle is meant the sides a and b, c being called the
Ć hypotenuse. The sides a and b are also called the legs of the triangle, particularly by early writers, since it was formerly the custom to represent the triangle as standing on the hypotenuse.
a b a b c The ratios
, and - have the following names : cb a' b
These definitions must be thoroughly learned, since they are the foundation upon which the whole science is built. The student should practice upon them, with the figure before him, until he can tell instantly what ratio is meant by sec A, cot A, sin A, and so on, in whatever order these functions are given.
There are also two other functions, rarely used at present. These are the versed sine A : 1 cos A, and the coversed sine A =1- sin A. These definitions need not be learned at this time, since they will be given again when the functions are met later in the work.