EXAMINATION PAPERS. +I. PREVIOUS EXAMINATION. December, 1886. PAPER I. I. DEFINE the secant and cotangent of an angle, and prove that (i) sec @ cosec =tan 0+cot 0, 1. From the definitions of the Trigonometrical Functions, prove that sin24=1-cos2A and sin A tan A=sec A-cos A. Prove that (cos + sin®4) - † (cos24 - sin2 A)2=. 2. Investigate the values of tan 45° and sin 60o. Two adjacent sides of a parallelogram are of lengths 15 and 24, and the angle between them is 60°; find the lengths of both diagonals. II. PREVIOUS EXAMINATION, CAMBRIDGE. June, 1887. PAPER I. 1. Define the sine and tangent of an angle, and shew how to find the sine and tangent whose cosine (m) is given. If sin Atan B, prove cos2A cos2 B=(cos B+ sin B) (cos B - sin B). †The Additional Subjects in the Cambridge Previous Examination, which are required of Candidates for any Tripos, are now (1887) either Mathematics, or French, or German. The Mathematical subjects are (i) The Trigonometry of one Angle, (ii) Elementary Dynamics [see Lock's Dynamics for Beginners], (iii) Elementary Statics. The Trigonometry is set in the first one or two questions in each of the two papers. The questions quoted above were set in December, 1886, and in June, 1887. 2. Trace the changes in the tangent of an angle as the angle changes from 180° to 270°. If sin 0=-, find tane; and explain by means of a figure the reason why there are two answers to the question. PAPER II. 1. Explain the mode of measuring angles in degrees, minutes and seconds. Find the number of seconds of angle through which the earth revolves about its axis in a second of time. 1. Prove that the angle subtended at the centre of a circle by an arc equal to the radius is the same for all circles. Express the angle as a fraction of a right angle. 2. Define the sine, cosine and tangent of an angle. Prove that these trigonometrical ratios are always the same for the same angle. Find these ratios for an angle of 45°. 3. Prove that cos (A+B)=cos A cos B-sin 4 sin B. Prove that the sum of the cosines of two angles is equal to twice the cosine of half their sum multiplied by the cosine of half their difference. 5. Prove that the logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers. Having given log 2=3010300, log 7=8450980, find the logarithms of (1·75), (24·5)—§. 7. Shew how to solve a triangle when one side and two angles are known. Find the side b in the triangle ABC from the following data: a=156.22, B=57°25', C=63°42′, log 15.622=1.1937366, L sin 57°25′=9·9256261, log 15.37552=1·1868297, L sin 58°53′ =9·9325330. 8. The angles of elevation of the top of a tower on a horizontal plane observed at two points distant a feet and b feet respectively from the base and in the same straight line with it are found to be complementary. Shew that the height of the tower is ab feet. If be the angle subtended at the top of the tower by the line joining the two points, prove that sin @= a~b a+b IV. OXFORD LOCAL EXAMINATIONS. 1. What is T? July, 1887. JUNIOR CANDIDATES. What is 'the angle whose circular measure is '? In a triangle ABC the angle A is x degrees, the angle B x grades, пх and the circular measure of C is ; find the number of degrees in each of the angles. 9 2. Prove that 2 sin= = ±√√1+sin A±√√1 -- sin A, and determine which are the correct signs when 270°>4>180°. 3. Obtain the following formula: (1) cos (4+B) = cos A cos B- sin A sin B; =tan p; (3) log 3 = 4771213, log 24668=4.3921339, log 11=1.0413927, log 24669=4.3921515: find the logarithm of 30-25 and calculate the value of {165 × (30) × √24}÷(121)7. 5. In a triangle ABC, a, b, c are the sides, s is the semi-perimeter, ▲ the area, R, r the radii of the circumscribed and inscribed circles: prove that (1) tan (2) sin B-C b -c cot (3) 2R=a cosec A=b cosec B=c cosec C ; (4) 2A=ca sin B. 6. In a plane triangle the sides a, b and the angle A are known; shew that in general two values of c can be found, and that the difference of these values is 2a2-b2 sin2 A. 7. A ladder placed at an angle of 75o just reaches the sill of a window 27 feet above the ground on one side of a street. On turning the ladder over without moving its foot, it is found that, when it rests against a wall on the other side of the street, it is at an angle of 15o. Find the breadth of the street. 8. If B=36° 46′, b = 311.8785, [L sin B=9-7771060, c=521.05, find C. log 31187=4.4939736, log 521.05=2.7168794, log 31188=4.4939875.] V. CAMBRIDGE LOCAL EXAMINATIONS. December, 1886. SENIOR STUDENTS. 1. Explain the method of measuring angles in circular measure, and find the circular measure of the angle of a regular pentagon. Prove that 3. Find an expression for all the angles which have a given sine. Solve the equations: (1) cos 0+tan 0=sec 0. (2) sin 0 - 2 sin 20 cos 0 + cos 30=cos 20. 4. Prove that in any triangle if a, b, c be the sides opposite to the angles A, B, C (1) (2) sin A sin B sin C where r and R are the radii of the inscribed and circumscribed circles respectively. If the line joining A to the centre of the inscribed circle meets the opposite side in D, prove that b+c A tan ADB=: tan -C *5. If is the circular measure of a positive angle less than a right angle, shew that sin lies between 0 and 03 when n is indefinitely in π 4 0=tan 0 - tan30 + ‡ tan5 0 — log (a+b√ √ − 1) = } log (a2 + ba) + √ = 1 tan-10). * See Higher Trigonometry. VI. OXFORD LOCAL EXAMINATIONS. July, 1887. SENIOR CANDIDATES. 1. Prove geometrically that the sine of 90°+A is equal to the cosine of 4 for all values of A. Find all the values of B, less than 180o, for which sin 5B = 3√√2. L. E. T. 20 |