cos me - Sm (-1)2 where the sum of the mth powers of the root of equation z2-1=0. 2" cos e cos 20 cos 22 0...cos 2"-10=1. (122) If a, b, c, are the roots of the equation then ... ... tan-1a + tan1b+tan-1c+...=tan-1 P1−P3+P5 − •• (123) Prove that e"+e"=2(1+22) {1+(3)2} {1+(3)2}... (124) The sum of the products of the reciprocals of the fourth powers of every positive integer is 3848 2 y (127) Prove that the coefficients of 62 and 04 in the expression vanish; explaining à priori why they do so. (128) Having given the formula (1 – 2266) (1 – 3243)... deduce the expression for sin 0 in factors. 322 2 (129) The coefficient of x" in the expansion of (130) By putting aia for in the expression of sin in (131) If a series of points are distributed symmetrically round the circumference of a circle, the sum of the squares of their distances from a point on the circumference is twice that from the centre. (132) If A1, A2, A3, ... A2n+1 are angular points of a regular polygon inscribed in a circle and O any point in the circumference between A1 and A2n+1, then the sum of the lengths = the sum 042+0A,+0A,+...+OAq (133) If from a point P straight lines PB1, PB2,...PB, be drawn to the middle points of the sides of a closed polygon A1A... A, and if the angles PB11, PB,A2, ... PB„A, be denoted by a, a, ... a, respectively, and the triangles PA,42, PAA3, ... PÄÂ1⁄2 by ▲1, ▲1⁄2, ..... A„, prove that A1 cot a1+A, cot a2+ .+A, cot a,=0. EXAMINATION PAPERS. I. SANDHURST-FURTHER. Nov. 1882. 1. Name and define the trigonometrical ratios. Prove that sec2 A+cosec2 A=sec2 A cosec3 A. If the cosecant of an angle between 90° and 180° is secant? And if the cosine of an angle between 540° and 630o is – }, what is the cosecant? 3. In a plane triangle ABC prove that— i. tan A tan B tan C-tan A+ tan B+tan C. ii. a sin A+ b sin B+c sin C-2 (a cos A+B cos B+y cos C), where abc are the sides and aßy the perpendiculars let fall on them from the opposite angles respectively. 4. Prove that the area of a triangle =a2 sin 2B+ b2 sin 24 ; and if R, r are the radii of the circumscribing and inscribed circles 5. Given log 10791812 and log 23=3802112, find the value of √ (3·6)3 × 41⁄2÷84, the mantissae for 46929 and 46930 being 6714413 and 6714506. In a triangle ABC, b=14, c=11, A=60°; find the other angles, having given L tan 11° 44' 29" 9.31774. 6. A measured line is drawn from a point on a horizontal plane in a direction at right angles to the line joining that point to the base of a tower standing on the plane. The angles of elevation of the tower from the two ends of the measured line are 30° and 18°. Find the height of the tower in terms of l, the length of the measured line. II. CAMBRIDGE PREVIOUS EXAMINATION. Dec. 1883. 1. Define the cosecant and tangent of an angle. 2. A man wishes to measure the distance between two points A and B between which lies an obstacle. He therefore walks from A to Cin a direction at right angles to AB a distance of 50 yards. He now finds that he can walk directly from C to B and that CB makes an angle of 60° with AC. Find the distance from A to B. 4. Find a formula for all angles having the same tangent as a. Solve completely the equation tan2 = 1. 5. Shew that sin (x+y)=sin x cos y + cos x sin y. Prove that sin (a - ẞ) cos 2ẞ + cos (a – ß) sin 2ß = sin (ß − a) cos 2a + cos (ẞ − a) sin 2a. If 6. Prove that sinx, cosy, find sin (x+y). - cos A (1) sin 4 = ± Determine the sign of the radical in (1) when A lies between 360° and 720". 7. If a, b, c be the sides of a triangle ABC, shew that a2= b2+c2-2bc cos A. If ABC be an equilateral triangle each of whose sides is eight inches, and in BC a point P be taken three inches from B, shew that AP is seven inches. Find all the angles of a triangle whose sides are 13 ft., 14 ft., and 15 ft. in length, having given log 2=30103, log 3=4771213, log 7=8450980 and I tan 26° 33′ = 9.6986847, tabular difference for 1'=3159, L tan 33° 41'=9.8237981, tabular difference for 1'=2738. 9. A base line 400 feet in length is measured from the foot of a vertical tower and at the end of this line the angular elevation of the top of the tower is observed to be 26° 33′ 54′′; shew that the height of the tower is very nearly 200 feet. Refer to question 8 for the necessary logarithms. III. WOOLWICH-PRELIMINARY. June, 1882. 1. Prove that the angle subtended at the centre of a circle by an arc equal in length to its radius is an invariable angle. One angle of a triangle is 45o, and the circular measure of another is 11. Find the third, both in degrees, and in circular measure. 2. Define the secant of an angle, and shew how your definition applies to angles between 180° and 270°. If sec A-2, what two values between 0° and 360o may A have? |