§ 32. ANTI-TRIGONOMETRIC FUNCTIONS. If y is any trigonometric function of an angle x, then x is said to be the corresponding anti-trigonometric function of y. Thus, if y sin x, x is the anti-sine of y, or inverse sine of y. The anti-trigonometric functions of y are written Similarly These are read, the angle whose sine is y, etc. 90° = cos 10 The symbol -1 must not be confused with the exponent 1. Thus sin-1x is a very different expression from 1 " sin x which would be written (sin x). On the Continent of Europe mathematical writers employ the notation arc sin, arc cos, etc., for sin-1, cos-1, etc. But the latter symbols are most common in England and America. There is an important difference between the trigonometric and the anti-trigonometric functions. When an angle is given, its functions are all completely determined; but when one of the functions is given the angle may have any one of an indefinite number of values. Thus, if sin y=, y may be 30°, or 150°, or either of these increased or diminished by any integral multiple of 360° or 2, but cannot take any other values. Accordingly sin1-30° ±2n, or 150° ±2nя, where n is any positive integer. Similarly, tan-11=45° ±2nπ or 225°+2nπ; i.e., tan-11=45° ±nπ. Since one of the angles whose sine is x and one of the angles whose cosine is x together make 90°, and since similar relations hold for the tangent and cotangent, for the secant and cosecant, and for the versed sine and coversed sine, we have where it must be understood that each equation is true only for a particular choice of the various possible values of the functions. For example, if x is positive, and if the angles. are always taken in the first quadrant, the equations are correct. -1 EXERCISE XV. 1. Find all the values of the following functions: sin-1√3, tan-1√3, vers1, cos-1(√2), esc1(√√2), tan-1∞, sec-12, cos−1(— ± √3). -1 2. Prove that sin-(-x)=-sin-1x; cos-1(x)=π-cos1x. 8. If x=√, find all the values of sin¬1x+cos¬1x. 5 10. Find the value of sin (tan131⁄2). 11. Find the value of cot (2 sin-13). 12. Find the value of sin (tan-1+tan-1). CHAPTER IV. THE OBLIQUE TRIANGLE. § 33. LAW OF SINES. LET A, B, C denote the angles of a triangle ABC (Figs. 31 and 32), and a, b, c, respectively, the lengths of the opposite sides. Draw CD AB, and meeting AB (Fig. 31) or AB produced (Fig. 32) at D. Let CD=h. Therefore, whether h lies within or without the triangle, By drawing perpendiculars from the vertices A and B to the opposite sides we may obtain, in the same way, Hence the Law of Sines, which may be thus stated: The sides of a triangle are proportional to the sines of the opposite angles. If we regard these three equations as proportions, and take them by alternation, it will be evident that they may be written in the symmetrical form, NOTE. Each of these equal ratios has a simple geometrical meaning which will appear if the Law of Sines is proved as follows: Circumscribe a circle about the triangle ABC (Fig. 33), and draw the radii OA, OB, OC; =R sin A. ..BC or a= 2 R sin A. In like manner, b=2 R sin B, and c=2R sin C. C Ꭱ B α M FIG. 33. obtain Whence we That is: The ratio of any side of a triangle to the sine of the opposite angle is numerically equal to the diameter of the circumscribed circle. § 34. LAW OF COSINES. This law gives the value of one side of a triangle in terms of the other two sides and the angle included between them. 2 Therefore, in all cases, a2=h2+AD2+ c2-2c × AD. The three formulas have precisely the same form, and the law may be stated as follows: The square of any side of a triangle is equal to the sum of the squares of the other two sides, diminished by twice their product into the cosine of the included angle. |