IX. CHRIST'S CHURCH, OXFORD, ENTRANCE SCHOLARSHIPS. 1883. (ii) (cos A+ sin A)(cos 24 + sin 2A)(cos A – sin 3A) cos 24 cos 44. (iii) 26 (cos3A+ sin3A) = cos 84 +28 cos 44 + 35. 3. Shew that m log. -= 2 m-'n 3 5 ( + + (m + n)2 + + (m + n) + ...}. Having given that log, 3=1.0986, find the value of log10 3. 4. Eliminate a, ẞ from the equations X= =(a sin2a+b cos2 a) cos2 ß+c sin2 ß, 2= y=a cos2 a+b sin2 a, = (b− a) sin a cos a cos B. 5. If circles can be both described about, and inscribed in a quadrilateral, whose sides are a, b, c, d, and the angle between the diagonals 0, then 6. Solve a triangle, having given the base a, altitude h, and the difference of the angles of the base a. Account for the two values obtained for the vertical angle, and shew which of them is possible. 8. The triangle A'B'C' circumscribes the escribed circles of the plane triangle ABC; shew that 9. If K be the centre of the nine-point circle of the triangle ABC, then 4AK2=R2+b2+c2 - a2, where R is the radius of the circumscribing circle. 10. If cos (0+√√-1)=cos a + √-1 sin a, and a, 0, 4 are real, prove that tan20 - tan2 a=sin2 0 sec2 a, and find a relation between e and p. 11. Sum to infinity the series— (i) cos a tan - cos 3a tan3 + cos 5a tan5 p... X. CHRIST'S COLLEGE, CAMBRIDGE. ENTRANCE SCHOLARSHIP. 1878. 1. Find the general expression for all angles which have a given tangent or cotangent. Solve the equation sec2 +3 cosec2 0=8. 2. Prove geometrically the formula: (1) cos (A-B)=cos A cos B + sin A sin B. (2) sin A+ sin B = 2 sin § (A + B) cos § (A – B). Shew that cosec A+cosec (4+ }π) + cosec (A+π)=3 cosec 34. 3. If be the circular measure of an angle less than a right angle, prove that sin is less than 0, but greater than 0 – 103. 4. Prove that if a, ß, y are any three plane angles (cos a + cos B+ cos y) {cos 2a + cos 26+ cos 2y - cos (ẞ+y) – cos (y+a) · cos (a+ẞ) } − (sina + sin ß+ sin y) { sin 2a + sin 2ẞ + sin 27 − sin (ẞ+7) -sin(y+a) - sin(a+B)} = cos 3a+cos 38+ cos 37. 5. Shew that r=4R sin 4 sin B sin C, where R is the radius of the circumscribing circle and r of the inscribed circle of the triangle ABC. If A, A' be the areas of the two triangles, in the ambiguous case (given A, a, b), prove that the continued product of the inscribed and escribed radii to the side b is equal to ▲▲'. 6. State De Moivre's theorem, and prove that there are n values and no more for the expression and deduce the expression for sin 0 in factors. Shew that the sum of the series XI. ST JOHN'S COLLEGE, CAMBRIDGE. June Exam., 1879. 1. Explain the method of measuring angles by degrees, minutes, &c. The numerical measures of the angles A, B, C of a triangle when referred to units 1o, mo, no, respectively, are in arithmetical progression, and when referred to units po, qo, ro respectively, they are in geometrical progression. Find A, B, C. 2. Define the sine and cosine of an angle, and prove that sin2 A+ cos2 A=1. If cos2 A+ cos B=1=sin2 A+ sin B, find A and B. cos A+ cos B=2 cos † (A+B) cos (4 – B). Find and from the equations cos a {cos a + cos (a + 0)} = =cos B{cos ẞ+cos (B+¢)}, 5. If be the circular measure of an angle less than a right angle, prove that sin 0, 0, and tan 0, are in ascending order of magnitude. If the unit of measurement be a right angle, find the limit of tan - sin 0 ase is indefinitely diminished. 03 6. Expand log, (1+x) in a series of powers of x. Prove that 2 (cos4+cos3 4+ cos A......) = cos2 4 - sin2+(cos1‡▲ – sin1‡▲) +(cos64 - sino 14) +...... 7. In any triangle the sides are proportional to the sines of the angles opposite to them. Through the angular point C of a triangle ABC is drawn any line CMN on which are dropped perpendiculars AM, BN. Prove that MN=AM cot B~ BN cot A. 8. Express the sine and cosine of half the angle of a triangle in terms of the sides. If ABC, A'B'C' be two triangles, such that prove that tan1⁄4 tan '=tan B tan B′ = tan C tan C. 9. Give the formulæ for the solution of a triangle in which one angle and the containing sides are given. If C=44°, a=43 ft., b=11 ft., find A and B. Having given log 2=3010300, log 3=4771213, L tan 22o=9-6064096, L tan 34° 17'-9.8336109, L tan 34° 19' 9.8338823. 10. Enunciate and prove De Moivre's theorem. If n be equal to 3m±1, prove that 11. Find the sum of the following series, each to n terms: (1) cos a+ cos 3a+cos 5a+...... XII. ST JOHN'S COLLEGE, CAMBRIDGE. MINOR SCHOLARSHIP, 1881. 1. Shew that in the expression for tan 12 in terms of tan A we should à priori expect a double result. Find tan 112o. 30', 2. A triangle is such that the product of two sides is equal to the square on half the base: prove that the difference of the sides varies as the distance from the vertex to the middle point of the base. 3. (i) If x, y, z be any angles, prove that sin(x − y − z) sin § (y − 2) + sin † (x + y − 2) sin (y + z) = sin 3x sin y. (ii) Also if A, B, B be the angles of an isosceles triangle, 2 sin2 (A – B) (2 – cos A) = (sin2 A + 2 sin2 B) (1 − 8 cos A cos2 B). |