Thus to express the trigonometric functions of angles greater than 90° in terms of those of angles less than 90°, determine first the sign of the function to be so expressed, next subtract from the angle whatever multiple of 90 is neces sary to make it less than DO°. 11 an even multiple of 90° is subtracted the name of the original function is retained, but if an odd mul tiple is subtracted the original function is re placed by the co-named function. E.g. to ex From the definition of the trigonometric func tions, it is evident that they bear certain rela tions one to another. Some of the fundamental ones are, sinze cos'e = 1, sine cse 0 = 1, cos P see 0 = 1, tall 0 cot 0= 1, tan = sin 0/cos 0, 1 + 0 = 0, 1 -4- cote 0 = from which many others readily follow. Besides these relations existing between the functions of a single angle, there are those connecting the functions of several angles. Thus sin (A ± B) = sinAcosB ± cosAsinB, cos (A ± B) = eosAcosB sinAsinB,tan A ± tan B tan (A B) tan A tau cot A. cot B r 1 cut (A B)= cot A ± and sin 2 A = 2 sin A cos A, cos 2A = A 2 tan A tan 2A = and cot2A = cot' A 1 2 cot A . which are easily derived from the corresponding formulas for A B by putting A = B. Some of the formulas for functions of half an angle are : sin IA = cOs cos 1A 71: tan 1 ± cos A 1 1 cos A 1 + COS and cot IA = 1 + cot A By reapplying 1 cot A these formulas it is evident that functions of 3A, 4A nA may be expressed as functions of A and also as functions of various fractional parts of A.
To every function there is an inverse function or anti-function just as to every logarithm there is an antilogarithm. The formula to express the angle whose sine is x is = 0, read "0 is the angle whose tine is x." Similarly y 0 is read, "0 is the angle whose tangent is y," or "anti-tangent of y equals 0." All inverse
functions admit of translation into the direct formulas. Thus = 0 reduces to sing = x, and = y =0 to tan 0= y. All in verse or anti-functions can be expressed in series, as in the ease of the functions. (See SERIES.) The following will serve as examples: x 1 .... (2)-1) x -- 1 3 5 -}- + 0 4 or 72r4-1 (2r + 1) + x =T s ,2n-1 + 9 + The values of the func tions of certain angles may be calculated by ref erence to geometric fig ures, but the tables of such values for all angles have been calculated to a close degree of approxi mation by means of the trigonometric series. The equilateral triangle serves to exhibit the values of the functions of and If the side he taken as 1, the figure shows that sin = V3, = = V3, and so on.
Similarly the functions of as sin 1 = = may be obtained from the square.
The following problems will serve to illustrate the use of trigonometry in practical mensura tion: (1) Required the height of a hill above the horizontal plane of an observer, the distance of the observer from the point below the summit being 5000 feet and the angle of elevation 30'. The height of the hill, represented by BC in the figure. is given by the equation BC = 30' 5000 feet = 0.1853 X 5000 feet = 926.5 feet, 30' being taken from a table of natural tangents. (2) Required the distance between two points A and B separated by an impassable swamp, the line AC, as represented in the figure, being 15 chain lengths, the angle A 15', and the angle C 32'. The length of the line AB 32' is given by the equation AB = s i 47' 0.9365 = 28.5, the sines of the angles being 0.4881 taken from a table of natural sines. Therefore AB is 28.5 chain lengths.
From the expressions of siux, eosx (see