Plane Trigonometry

* PLANE TRIGONOMETRY All solutions of problems of distance and direction in a plane depend upon finding unknown parts of a triangle when three parts are known (or "given"). Most simply, these may be sides and angles, at least one being the length of one side. For cal culations involving the size of an angle, the ratios of sides in a right triangle are employed, a right triangle having one angle equal to the given angle, or differing from it by some even number of right angles; or the supplement of such an angle. For convenience of measurement and uniformity of language a technical system is adopted, as follows : The angle to be measured is depicted in a vertical plane, facing the observer. The angle is defined by one straight segment OA, called its initial side, and a second segment OB, its terminal side. It is often said to be generated by the continuous rotation of a straight segment about the point 0, from the position OA to the position OB. Since this rotation has two conceivable directions or senses, one is called the negative sense, namely, that performed by the hands of a clock as seen by an observer standing before its face ; the opposite sense of rotation is called positive. If one could stand at the position of the north pole-star and face toward the sun, the planets of our solar system would appear in their orbits to revolve in the positive sense about the sun.

Fix arbitrarily a point

D on the terminal segment (fig. I), and let d denote the distance OD. Draw a line OC perpendicular to OA and measure the Cartesian coordinates x and y of D, taking as x the distance of D from the line OC (right or left), and y its distance above or below the initial line or that line produced in the direction AO. The indefinitely extended lines OA and OC are termed the axes of coordinates (see COORDINATES) in this diagram, OA being the x axis and OC the y axis; so that x coordinates are parallel to the x axis, y coordinates parallel to the y axis. Co ordinates are measured from an axis, to the point D, and are positive for an acute angle. Hence on the diagram which is used here, a positive x extends from the y axis toward the ob server's right, a negative x toward his left ; a positive y upward from the x axis, a negative y downward. These conventional definitions enable us to characterize the picture of any real angle by three real numbers, d, x and y. But any new selection of the point D on the terminal line, as D', would give a second set of numbers, d', x', y', which are proportional to d, x and y. Accord

ingly not these numbers, but their ratios, are chosen to describe or measure the angle.

Trigonometric Functions or Ratios, of an Angle.—Six names are assigned to the six ratios of the numbers, x, y, d. Some symbol, as K, is used for the angle. Then y/d is called the sine of K, or sin K = y/d x/d „ „ „ cosine of K, or cos K=x/d y/x „ „ „ tangent of K, or tan K=y/x x/y „ „ „ cotangent of K, or cot K=x/y d/x ff fl secant of K, or sec K=d/x d/y „ „ „ cosecant of K, or cosec K=d/y By definition three of these six are reciprocals of three others; cot K= 1/tan K, sec K=1/cos K, cosec K= 1/sin K.

The sine is also the product of two others; sin K= cos K.tan K. Since also the Pythagorean relation connects x, y and d,—viz., are quadratic relations or identities among the six ratios. One of these is = I. By the use of these five relations or any equivalent set, when the numerical value of any one ratio or function is known the five others can be found numerically, but not always their plus or minus sign. If however the sign of a second ratio is given, a ratio not the reciprocal of the first, then the rest can be found completely.

Limits of Values of the Six Functions.—From the fact that d is numerically greater than either x or y, and that either one of the latter may become equal to zero for some angle which is a whole number of right angles, it is seen that two functions of real angles have outer boundaries, two have inner boundaries, and two are unbounded.

tan

K and cot K are unlimited in value.

Unit of Real and Imaginary Angles.

In geometry the unit of angle, a degree, was defined as one ninetieth part of a right angle, a degree is divided into sixty equal parts called minutes, and each minute into sixty equal parts called seconds. In terms of these units an angle can be described by three whole numbers if it does not demand more precise measurement, or it may be de scribed by the aid of a decimal fraction of a unit. Astronomers use degrees, minutes, seconds, and often tenths and hundredths of a second. Land surveyors use degrees and minutes, and some times tenths of a minute. For purely theoretical discussions a larger unit is usually employed. This is the radian and is best described by saying that 7r radians are the same as 18o degrees. Using the symbols " for degrees, minutes, seconds, we may say that