TRIGrONOMETRY. 462 TRIGONOMETRY. these relations existing between the functions of a single angle, there are those connecting the functions of several angles. Thus sin (A ± B) =: sinAeosB ± cosAsinB, cos (A ± B) = cosAcosB =F sinAsinB, tan A ± tan B tan (A ± B) = i ^ ^n A tan B' cot A cot B =F 1 cot (A ± B) = cot A ± cot B = and sin 2 A = 2 sin A cos A, cos 2A = cos' A — 2 tan A sin^A, tan 2A = :i . . ■ . and cot2A = cot^ A — 1 n . . — . -wnich are easily derived from the corresponding formulas for A -j- B by putting A = B. Some of the formulas for functions of half an angle are : sin lA = ± • -4 1 + cos A tan JA = 1- -cos A cos 1 A 2 V^ — COS A A 1 4- COS A' and cot J A = ^.V4 + cot A By reapplying 2-4. -Jr+l
- +•
+ (2r + 1) 3.20-1 + (-!)" 2-^1 The values of the func- tions of certain angles niaj' 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 G0° and 30°. If the side be taken as 1, the figure shows that sin 60° = 1/2 V3, coseO" = %, tan60° =: f3, and so on. Similarly the functions of 45 1 tan~i X :=~r — -K-i-^ — . . v/2 square. as sin 45°=cos 45° % V 2, may be obtained from the The following problems will serve to illustrate the use of trigonometry in practical mensura- tion: (1) Required the height of a hill above ■ 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 ever.v logarithm there is an antilogarithm. The formula to express the afigle whose sine is x is sin~' a; = 6, read " 6 is the angle whose sine is x." Similarly tan~' y = 6 is read, " 8 is the angle whose tangent is y," or "anti-tangent of y equals $." All inverse functions admit of translation into the direct formulas. Thus sin~'iB = 6 reduces to sine = X, and tan-^ z^ y ^6 to tan 9^ y. All in- verse or anti-functions can be expressed in series, as in the case of the functions. (See Sebies. ) The following will serve as examples: X 1 a? _ 1-.3 a° l-3....(2r— 1) sin 'x=:j- + ^o- + rr:Ts+- ■ the horizontal plane of an observer, the distance of the observer from the point below the summit lieing 5000 feet and the angle of elevation 10° 30'. The height of the hill, represented by BC in the figure, is given by the equation BC = tanl0° 30' 5000 feet = O.l'sSS X 5000 feet = 926.5 feet, tanlO° 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 40° 15', and the angle C 110° 32'. The length of the line AB . . , ,, ,. .^ ACsinllO° 32' is given by the equation AB = 15-0.0365 sinl50° 47' „ .,, = 28.5, the sines of the angles being 0.4SS1 " taken from a table of natural sines. Therefore AB is 28.5 chain lengths. From the expressions of e", sino!, cosa; (see Series ) , it follows that e'" = cosx -{- i sinj; (see CosiPLEX Number), and that e~'" = cosa; — i sina; ; whence by adding and subtracting,
- ^ (.-is
, sino! -. Similarlv 2 ' 2i the other functions may be expressed in terms of c", e~'^. These are the exponential expres- sions for the circular functions of x. If » is omitted from these exponentials, the resulting functions are called the hyperbolic cosine, hyper- bolic sine of the angle x. Hyperbolic functions are so called because they have geometric rela- tions with the equilateral hyperbola analogous to those between the circular functions and the circle. The common notation for such functions is sin/jS, cos/j^, tan/jS, corresponding to the circular function sinfl, cosff, tanff. The values of these functions have been tabulated and are of service in analytic trigonometrj-.
