PLANE TRIGONOMETRY. 





, on a = 53, c 47, C 30* 42* 0". Find A, B, and b (uraiM A 

 = 199. and B <B 21' 20". Find A (obtuse), C. 



von a 100, o 62, and C - 33 19'. Fin 1 L, B, ana b. 

 r/i o;if MI!-', K, anJ two a>iglea, A and B. Find C, I, 



180-(A+B)=C. 



By (65), i-!". 

 a sin. A 



.. log. 6 = log. o + log. ain. B log. sin. A. 

 Similarly, log. c = log. a + log. sin. C log. sin. A. 



EXERCISE G. 



1. Given a ^ 217. B = 50 21' 33". C = 02 41' 20". Find A, b, and c. 

 .von a = 1000, B = 123* 15' 15", C = 36' 5^. Find A, t, and c. 



\\ll. To find the Area of a Triangle. There are two useful 

 .u for finding the area one in terms of any two eidos 

 included angle, the other in terms of the throe side.-s. 



1. Area = J be sin. A. 

 If A is a right angle, area evidently = hv1f the rectangle 



Fig. 17. 



under the adjacent sides, which agrees with the statement, since 

 sin. 90 = 1. 



If A is acute, as in Fig. 15, drop c P perpendicular to A B, or 

 A B produced (Fig. 1C). Then, by Euclid II. 1, 

 Area = J A B X c P. 



C P 



But A B = c, and since = sin. A, c P = b sin. A ; 

 o 



therefore area = $ be sin. A. 

 If A be obtuse (Fig. 17), drop c P as before, on B A produced ; 



then area = A B X c P, 



But A B = c and c P = c A sin. CAP, and sin. c A P = sup- 

 plement of A. 



.-. Area be sin. A ........................ (77) 



Or, log. 2 area = log. b + log. c X log. sin. A - 10. 

 2. Substituting in (77) the value of sin. A given in (73), we 

 ct Area= Vs (s - a) (s - b) (s - c) ............ (78) 



^ lo g' s + lo S" ( g " a ) + lp g- ( 8 - *>) + log- (s - c) . 



bat it is often easier to work out (77) arithmetically than to 

 employ logarithms. 



EXERCISE 7. 



1. Given b = 35 feet, c = 117 feet, and A = 27. Find the area. 



2. Given a = 1000 yards, b = 2'5 miles, and C = 42. Find the area. 



3. Given 6 = 2'3U, c = 1'5:J7, and A = 49* 6' 20". Find the area. 



4. Given a = 287'1, e = 310 25, and B = 114 23* 32". Find the area. 



5. Given a = 49, b = 98, c = 53. Find the area without employing 

 logarithms. 



C. Given a = 603, b = 507, c = 721. Find the area. 



7. Given a = 0'45, b = 0'3t, c = 0'23. Find the area. 



8. Given o = 2'05, 6 = 1'67, c = 27. Find the area. 



We have now concluded our investigation of theoretic Trigo- 

 nometry, or rather of such parts of the theory as will enable us 

 to apply our knowledge largely in practice. There are formula; 

 for other ratios or values, such as for the radius of the circle in- 

 scribed in, or circumscribed about, a given triangle, the area of 

 the circumscribed circle in terms of the sides, the area of any 

 polygon inscribed in a circle (whence the area of the circle 

 itself may be obtained approximately), and the like ; but those, 

 although useful, are not needed to enable us merely to solve 

 ' heights and distances," upon which the practical art of sur- 

 voying mainly depends. A complete survey of a coast or 



country may bo made, and heights and distances accurately 

 calculated, without a single actual measurement being taken, 

 except one at starting called the late-line. (It U usual, 

 however, to check a rcimlt here and there by actual measure- 

 ment.) By choosing or marking spot* or object* at convenient 

 distances apart, the whole district is divided into triangles, 

 and it U obvious that a knowledge of one side of the first tri- 

 angle calculated (the base-lino before mentioned), an instrument 

 for measuring angles, and a level, are all that are required to 

 enable it to be completely surveyed. 



A few cases will bo given in the next lesson as illustrations 

 of the commonest practical uses of Trigonometry, and the Key 

 to Exercises 2 to 7 inclusive. 



TERMS USED IN COMMERCE. VII. 



PAR OF EXCHANGE. The comparative intrinsic value of the 

 specie of different countries according to their fixed standards 

 of weight and purity. 



PARTNERSHIP. The combination of two or more individuals 

 for the purposes of business in common, each deriving a share 

 of the profits, or bearing a corresponding share of the losses 

 arising from it. 



PASS BOOK. A book passing between bankers and their 

 customers, which records all payments and receipts. 



PASSPORT. A document granted by a consul, giving a de- 

 scription of the owner, and entitling him to pass through or to 

 reside for a time in the country for which it is given. In mari- 

 time law, a document carried in time of war by a vessel to prove 

 her nationality. 



PATENT LETTERS PATENT. A privilege granted under the 

 Crown seal, conveying to the persons specified the sole right to 

 make use of some new invention or discovery therein stated. 



PENALTY. A sum to be forfeited for the non-completion of 

 a contract or for a part of it. 



PER CENT. "By the Hundred." Thus 5 per cent, would be 

 five out of every hundred. 



PERMIT. A licence from the Excise authorities permitting 

 the removal of goods upon which duty has been paid. 



PILOT. A person duly qualified and authorised to conduct 

 ships through rivers, into or out of port, or through certain 

 channels or roads. 



PLANT. A trade term comprehending fixed machinery, imple- 

 ments, or other requisites for carrying on a business. 



POLICY OF INSURANCE. A document by which insurance 

 companies and underwriters secure to the parties contracting 

 with them for life, fire, marine, or accident insurance, an in- 

 demnity against loss from the risk incurred. It is a document 

 of considerable importance ; stating the names of the insurers 

 and of the insured, the amount and exact nature of the in- 

 demnity, and of tha risk incurred. 



POST, To (Book-keeping). To transfer an entry from one 

 book to another. 



POSTDATE. To date a letter or document of any description 

 later than the day on which it is written. 



POST OBIT BOND. A bond, the main condition of which is 

 that it only becomes payable after the death of some person 

 whose name is therein specified. 



PRECIS-WRITING. Writing the contents of a document in as 

 short and condensed a style as possible. 



PREMIUM. An additional sum beyond a standard or fixed 

 price. 



PREMIUM (INSURANCE). The per-centage or sum paid by 

 the insured for the indemnification granted by the insurer. 



PRESENTMENT OF A BILL. The act of demanding, or pre- 

 senting for, acceptance or payment. 



PRICE CURRENT. A list or enumeration of various articles 

 of commerce, with the market price of each. 



PRIMAGE, OR HAT MONEY. A customary per-oentage paid 

 by shippers, in addition to the freight of goods, and considered 

 to be for the master of the vessel, for his care and trouble in 

 taking charge of such goods while on board. 



PRIME COST. The first cost, before charges begin to accrue. 



PRINCIPAL signifies the responsible person. It also applies 

 to the partners in any establishment, who are spoken of as 

 the principals. In Banking, the sum on which the interest 

 arises. 



