Plane Trigonometry

By reference to Fig. 6, it is seen that sin a = ; similarly, sin /3 =— r 2 and sin y = ; whence, for 2 any triangle of sides a, b, c, and angles a, A, y, there holds the relation: = — —.2r, sine sin sin y r being the radius of the circumscribed circle. This relation is the LAW OF SINES. It is plain, Fig. 7, that a = b cos y+ c cos whence c cos 13= a — b cos y ; squaring, c' cos' /3= a'+ b' co? cos y; by law of sines, c sin /3= b sin y; squaring this, adding to the preceding equations, and noting that sin'+cos'= 1, there results c'= + b'-2ab cos 7. This and the analogous equations, b'= c'+ a'-2ca cos /3, a'= c'— 2bc cos a, similarly obtain able, together constitute the LAW OF COSINES. The LAW OF TANGENTS, deduced at a later stage of this article, is, a + b: a —b = tan f(afi): tan 4(a — ft), where a and b are any two sides of any triangle, and a and /3 are the opposite corresponding angles.

Addition These serve to ex press functions of an angle in terms of the like or unlike functions of its parts. Let OP = OP= r = r in Fig. 8; a' and a any two angles.

It is evident that x = r cos a, x'= r cos a', y = r sin a, 3(= r sin a'. Also, by the Pythagorean theorem and the law of cosines, d' = (x (y—Y cos (a' — a), whence, on substituting r cos a, etc., and reducing, it is sound that cos (a' — .m cos a' cos a + gin a' sin a. The four formula (1) cos(a — ,8)= cos a cos t9 + sin a sin /9.

(2) cos (a + ,3) = cos a cos /3 — sin a sin ft.

(3) sin (a + p) = sin a cos /3 + cos a sin ft.

(4) sin (a — M = sin a cos — cos a sin /3, are together named the Addition Theorem for Sine and Cosine. Formula (1) is equivalent to the preceding one; (2) results from (1) on re placing — IT for 13; (3) from (1) on writing 2 Tr — —a for a; and (4) from (3), as (2) from (1). Dividing (4) by (1), and both terms of the resulting right-hand quotient by cos a cos /3, there results: tan (a —13) = (tan a — tan /3): (1 + tan a tan 13), which with the relation, tan (a + ft) r..--_ (tan a + tan r3) : (1 — tan a tan 13), constitutes the Addition Theorem for the Tangent. The analogous relations for the re mauling functions are omitted as being but little used.

Prosthapheretic From the ad dition theorems, which concern functions of angle sums and differences, may be easily de duced equally important formula concerning function sums and differences. Replacing a +P by u and a ---,3 by v in (1), (2), (3), (4), adding (3) and (4), then (1) and (2), and then adding or subtracting one obtains the four formula: g? sin : sin = ; ens +( ]u 7) sin u — sin v = 2 cos 11(u + v) sin +)(u — v), (8) cos u—cos v =— 2 sin Ru + v) sin }(u—v). These relations, which have been named bros thapheretic,* express sums and differences in terms of products, and so render them suitable for logarithmic computation.

Some Impoitant Deductions from Fore going Formulae.—Setting in (2) and (3), there result (9) cos 2a = cos'a — sin'a, (10) sin 2a= 2 sine cos a.

The former combined with the relation 1 = sin' a+ cos'a yields (11) 1 cos 2a =2 cos'a, (12) 1 — cos 2a= 2 sin's.

Again, putting a=/3 in addition theorem of tangent, the result is tan 2a = 2 tan a: — tan'a). Division of (12) by (11) yields (1 — cos 2a) : (1 + cos 2a) = tan'a, whence follow 1 — tan'a (13) cos 2a = — 1 + 2 tan a (14) sin 2a = I + tan' a' .

Once more by definition tan a = sin a :cos a. Squaring each member and adding 1 to each square, one finds the most used of secant for mula, sec'a= 1 + tan'a. The corresponding co secant relation is coseea=l+ cot'a. Hosts of other more or less useful and interesting kindred formula, readily deducible, may be found in the current textl?ooks and in pocket manuals for engineers. This paragraph will be closed with a deduction of the above presented Law of Tangents. By the Law of Sines, a :b = sin a: sin ft; whence, by "composition and division' (a + b) :(a — b) = (sin a + sin /3): (sin a— sin t3); expanding the right hand number by (4) and (3), and applying the definition of tangent, the relation sought is found to be (a + b) : (a — b) = tan *(a P):(tall_f Solution of Etymologically trigonometry is triangle measurement, and, though the science wonderfully exceeds the verbal significance of its name, yet measure ment of triangles is a very important, and, at the same time, the most generally familiar, one of its manifold applications. A triangle is determined by three independent data, of which the simplest are: two sides and an angle; two angles and a side; three sides. The three angles are not independent, any pair of them determine the third angle. Let a, b, c denote the lengths of the sides, and a, 13, the corresponding (opposite) angles, of any triangle. The data being those mentioned, three cases arise: (i) given a pair of opposite parts, and one other, as a, a and b, or a, a and 19, to find the remaining parts; (ii) given three adjacent parts, as a, y, and c, or a, c and to find the rest; (iii) given three alternate parts, a, b and c, to find the angles. In case (i), it is sufficient to employ the Law of Sines; in (ii) the Law of Tangents; in (iii), the Law of Cosines. In (i), if the 'cone other° part be a side, as b, the Sine Law yields the sine of p, the opposite angle. But, as sin fi sin (7r— /3), the problem presents an ambiguity, which, in every actual example, is readily resoluble by easy considerations explained in every text book of trigonometry. The Cosine Law is equivalent to the equation: cos a = e—a'):2bc. The numerator not being a product, the formula is not adapted to loga rithmic use. From it, however, is readily derived an adequate formula which is so adapted. It is (15) tan in = (s —a)(s— c), s a where s is f(a + b c). Similar relations hold for fJ and y. The significance of (15) and its elements is further exhibited by Fig.