and (ii) If 2 cos a cos ẞ+x (sin a+ sin 3)+1=0) prove that x2 cos y cos a +x (sin y + sin a)+1=0. 5. Prove that the distance between the centre of the inscribed circle and the intersection of perpendiculars from the angular points on the opposite sides of a triangle is 2R {vers A vers B vers C-cos A cos B cos C}, where R is the radius of the circumscribed circle. P 6. Prove that {cos +√(−1) sin e}" admits of no more than g values. Find the continued product of the 4 values of {cos +(-1) sin. XIII. CLARE, CAIUS, AND KING'S COLLEGES. June Exam., 1880. 1. Draw a curve representing the change in sign and magnitude of tan 20 while changes from 0 to π. If 3 (1+tan2 A tan2 B) +8 tan A tan B = tan2 A+tan2 B, A and B differ by some multiple of . 3. If sin 3A be given, and from this value tan A is to be found, shew à priori that six values are to be generally expected. Prove by help of this, or otherwise, that tan2a{tan2 (}π − a) +tan2 (}π+a)} +tan2 (§π − a) tan2 (}π+a) =6 sec2 3a +3. {4 cos (x+a) -1}{4 cos (x − a) -1}=5 (2 cos 2a - 1) ......(2). Eliminate a from the equations 5. State the principle of proportional parts in the use of tables of functions. What is meant by saying that the differences are (1) insensible, (2) irregular? Prove that they are both insensible and irregular in the case of the logarithmic sine when the angle approaches. Determine a limit to the error which can be made in finding the α 100 logarithm of N+ from seven-figure tables from those of N and N+1, where a lies between 0 and 100 and N consists of 5 digits. 6. Explain fully the method of solving a triangle, given two sides, the included angle and a table of logarithms. ABC, AB'C' are two triangles having AB, BC equal respectively to AB', B'C', and A, C, C' are collinear. If the angle BAB' is 1", find correctly to a tenth of a second the angle between BC and B'C', where AB=2BC and ▲ ABC=60o. 7. ABC is a triangle and tangents are drawn to the nine-point and circumscribing circles at the four points where the perpendicular from A on the opposite side BC meets them. Prove that the four tangents form a parallelogram of area 8. Find the limit of the expression (90-0) tan 6o as 6 approaches 1 9. Given that xm + = хт 2 cos me for all values from 0 to m, shew that the formula holds when m+1 is written for m. Deduce the and deduce the exponential value of sin 0. Shew that sin1 (cosec 0) = (2λ+1) π + √( − 1) le cot § (\π+0), where X is any integer positive or negative. 11. Assuming the factorial expressions for sin @ and cos 0, prove that tan 0>0, provided ◊ lie between 0 and 1. By means of the result in question 8, or otherwise, prove that the infinite product sin a+3 sin 2a +5 sin 3a + ... to n terms.........(2). XIV. CHRIST'S, EMMANUEL, AND SIDNEY SUSSEX COLLEGES. June Examination, 1882. 1. Define the cosine and the tangent of an angle. Trace the changes in sign and magnitude of (1) tan 0, √/3+tan 0 (2) √3-tan o' as varies from 0° to 360o. 2. Prove geometrically that sin A - sin B = 2 sin § (4 – B) sin § (A+B). Shew that (1) cos 2a cos2 (B+y) + cos 2ẞ cos2 (y+a) + cos 2y cos2 (a+B) In order to ascertain the distance of an inaccessible object P, a person measures a length AB=20 yards in a convenient direction; at A he observes that the angle PAB=60o, and at B that the angle PBA=119o 20'. Find approximately the distance BP. To what degree of accuracy is your result correct, supposing (1) that there is no error in the measurement of the angles, (2) that there is an error of 1' in the measurement of each angle? 4. In any triangle ABC, shew that and that a=b cos c+c cos B, a2=b2+c2-2bc cos A. If N be the foot of the perpendicular from C on AB, and the circle on CN as diameter cut CA, CB in P and Q respectively, shew that the angle BPN is equal to the angle AQN. 5. Express the area of a triangle in terms of its sides. A straight line AB is divided at C into two parts of lengths 2a and 26 respectively. On AC, CB and AB as diameters semicircles are described so as to be on the same side of AB. If O be the centre of the circle which touches each of the three semicircles, shew that its radius = ab (a+b) a2+ab+b2' and that its diameter is equal to the altitude of the triangle AOB. 6. Shew how to find the height and distance of an inaccessible object on a horizontal plane. A person wishing to ascertain the height of a tower stations himself in a horizontal plane through the base at a point at which the elevation at the top is 30o. On walking a distance a in a certain direction he finds that the elevation of the top is the same as before, and on walking a distance five-thirds of a at right angles to his previous direction, he finds that the elevation of the top is 60o. Shew that the height of the tower is a ora. Explain the two results. 7. In a triangle ABC, I, I' and O are the centres of the inscribed circle, the escribed circle opposite A and the circumscribing circle respectively, and R is the radius of the latter circle. Shew that (1) 012=R2 (1 – 8 sin 4 sin B sin (C), 8. Explain the meanings of sin-1x and tan-1x. How many bounding lines are required to construct all the angles included in the formula sin-1a+cos-1b+tan-1c? Shew that sin-1a+cos-1b=sin-1 (ab + √1 − a2 √1 − b3). sin-1 2(x+y) (1-xy) (1+x) (1+ y2) (y+2) (1-yz) + sin-1 (1+ y2) (1 + z2) +sin−1 (≈+x) (1 − 2x) =2π. 9. Assuming De Moivre's Theorem find the expansions of sin no and cos no as homogeneous functions of sin 0 and cos 0. 10. Investigate Gregory's series for the expansion of tan-1x in |