same as the number of parts into which it is required to divide AB). Let aD be the last of the parts set off. Join BD, and through a, a', &c. draw ab, a'b', &c. parallel to BD, then it may be readily proved that AB is divided as required in the points b, b', &c. 7. The rectangle contained by the diagonals of a quadrilateral figure inscribed in a circle, is equal to both the rectangles contained by its opposite sides. Euclid, Book VI., Prop. D. 8. Define the tangent of an angle, find the value of tan 60°, and prove that tan (180° - A)=-tan A, and tan (90°+4)= cot A. Also trace the changes in sign and magnitude, as A varies from 0° to 180°. Todhunter, Trig., Arts. 26, 37, 48, 52, 58. 9. If A and B be each less than a right angle, and A> B, prove that cos (AB) = cos A cos B+ sin A sin B. Draw the figure (1) when A is greater than a right angle, (2) when A is <B. First part of question, Todhunter, Trig., Art. 77, for second part, see figs. 21a, 21b. 10. If the sine of an angle is given, shew how to find the sine of half the angle, and explain why the general expression obtained ought to include four values. Given sin 300° = √(3), find sin 150°. : Tod. Trig., Arts. 98, 99. 2 sin 150° = √(1 − sin 300°) + √√(1 + sin 300°) Solve the equation sin x cos x + a sin2x = b. Let tan a = a, then equation becomes 12. The angles in one regular polygon are twice as many as in another regular polygon, and an angle of the former is to an angle of the latter as 3:2. Find the number of sides. Let 2n be the number of sides in the first polygon, whence n=4; therefore the number of sides required is 4 and 8. 13. În an oblique-angled triangle, given a=145, b=178, B=41° 10', find A. log 178 = 2.2511513, log 1452.1613680, = log sin 41° 10' 9.8183919, log sin 32° 21′ 54′′ = 9.7286086, and shew whether there is any ambiguity in the result. Result A = 32° 21′ 54′′. There will be no ambiguity in this case, since B is an acute angle and b>a. 14. In any triangle given a, b, and 4, find an equation for determining c; and discuss its values according as a is equal to, less than, or greater than b sin A. Tod. Trig., Art. 235. K ROYAL CIVIL ENGINEERING Papers Set for Admission to. ARITHMETIC AND MENSURATION. WEDNESDAY, 20TH JUNE, 1877. 2P.M. TO 5 P.M. 1. Add together 7, 2, 4, 71, and f. Ans. 181. 2. Subtract 8 from 94. Ans. 176. 3. Multiply together 21, 1, 13, %, and §. Ans. 1. 4. Divide 8 by 7. Ans. 12. 5. Add together 25-069, 50462, 1714-8342, and ·0890653. Ans. 1740 4968853. 6. Subtract 594-1307 from 601.05. Ans. 6.9193. 7. Multiply 540.5 by 00169. Ans. ·913445. 8. Divide 33736854 by 25.8. Ans. 0130763. 9. Express 01573 of a guinea in pence and the decimal of a penny. Ans. 3.96396 pence. 10. Add together 7, 1, 2, and 5. Ans. 15. ARITHMETIC AND MENSURATION. 11. Subtract 7113 from 1004. Ans. 285. 9 33 67 15 12. Multiply together 4, 2, 21, 83, and 45. Ans. 12. 13. Divide 131 by 4. Ans. 31. 14. Add together 3.075 of a cwt. and 4.35 of a lb., and give the answer in ounces and the decimal fraction of an ounce. Ans. 5580 oz. 15. Subtract 10-056 of a yard from 1201 of a furlong. Ans. 254.164 yards. 16. Multiply 056379 by 000635. Ans. 000035800665. 17. Divide 64.95 by 158 to three places of decimals. Ans. 411.075. 18. Express 59685 acres in roods, poles, yards, &c. Ans. 2r. 15p. 15yds. 5 in. 19. The net rental of an estate after deducting 7d. in the £ for income-tax and 5 per cent. on the remainder for the expenses of collecting is £959. 3s. 8d., find the gross rental. Ans. £1040. 20. Find by duodecimals the cost of the lead lining of a rectangular cistern 3 ft. 2 in. long, 2 ft. 8 in. wide, and 2 ft. 6 in. deep, at 3d. per lb.; 7 lbs. of lead covering a foot of surface. Ans. £3. 5s. 9ğd. 21. Find the surface of a globe of 24 inches diameter. Ans. 1809-5616 square inches. |