plane of the objects is known to be 517.3 yds., angle ACB is found to be 15° 13' 15". The angles of elevation of C viewed from A and B are 21° 9' 18" and 23° 15' 34" respectively. Find the distance from A to B. III. (Cornell, September, 1891.) 1. Trace the value of tan ✪ and that of csc 0, as ◊ increases from 0° to 360°. 4. Derive an expression for the sine of half an angle in a triangle in terms of the sides of the triangle. 5. Construct a figure and explain fully (giving formula) how you would find the height above its base, and the distance from the observer, of an inaccessible vertical object that is visible from two points whose distance apart is known, and which can be seen from one another. 6. Given two sides of a plane triangle equal respectively to 121.34 and 216.7, and the included angle 47° 21' 11", to find the remaining parts of the triangle. 7. In a right triangle, if the difference of the base and the perpendicular is 12 yds., and the angle at the base is 38° 1' 8", what is the length of the hypotenuse? IV. (Cornell, June, 1892.) 1. By means of an equilateral triangle, one of whose angles is bisected, find the numerical values of the functions of 30° and 60°. 2. If be any angle, prove that sin 0=tan 0 : √1+tan20, cos 0 = √csc2 0 — 1 : csc 0. 4. Find sin 2 0, cos 2 0, and tan 2 0, in terms of functions of 0. 5. Assuming the law of sines for a plane triangle, prove that (a+b): c = cos(A-B): sin C, (a—b) : c = sin † (A — B) : cos † C. 6. At 120 feet distance, and on a level with the foot of a steeple, the angle of elevation of the top is 62° 27'; find the height. 7. Solve the plane triangle given the three sides, a=48.76, b=62.92, c=80.24. V. (Harvard, June, 1889.) 1. In how many years will a sum of money double itself at 4 per cent., interest being compounded semi-annually? 1+√1 2. Given sin2x , find sin 2x and tan 2 x. 2 3. Find all values of x, under 360°, which satisfy the equation √8 cos 2 x=1-2 sin x. 4. What is always the value of 2 sin2x sin3y+2 cos2x cos2y - cos 2x cos 2y? 5. Find the area of a parallelogram, if its diagonals are 2 and 3, and intersect each other at an angle of 35o. 6. Find the bearing and distance from Cape Horn (55° 55' S., 67° 40′ W.) to Falkland Island (51° 40' S., 59° W.). VI. (Harvard, June, 1890.) 1. In a certain system of logarithms 1.25 is the logarithm of. What is the base? Be careful to remember what 1.25 means. 2. Find the tangent of 3x in terms of the tangent of x. 3. One angle of a triangle is 35°, and one of the sides including this angle is 24. What are the smallest values the other sides can have? 4. Find all values of x, under 360°, which satisfy the equation tan 2x (tan x-1)=2 sec2x-6. 5. Two ships leave Cape Cod (42° N., 70° W.), one sailing E., the other sailing N.E. How many miles must each sail to reach longitude 65° W.? 6. If A+B+C=180°, find the value of tan A+tan B+tan C-tan A tan B tan C. VII. (Harvard, September, 1891.) 1. What is the base, when log 0.008 - 1.5? 2. If cos (a—b) = 3 cos (a+b), find the value of 3. The area of an oblique-angled triangle is 50. is 30°, and a side adjacent to that angle is 12. triangle. sec (a+b). sec a sec b One angle 4. Find all values of x, less than 360°, which satisfy the equation 5. Find, by Middle Latitude Sailing, the course and the distance from Cape Cod (Lat. 42° 2' N., Long. 70° 4' W.) to Fayal (Lat. 38° 32' N., Long. 28° 39' W.). 6. In any triangle ABC, prove tan A, tan B+ tan 1⁄2 4 tan C+tan B tan C1. VIII. (Harvard, September, 1892.) (Take the questions in any order. One of the starred questions may be omitted.) 1. What is the base of a system of logarithms in which log (243)=2.33}? *2. Given the area of a right triangle, and the smallest angle, find the legs of the triangle in terms of the data. 4. One angle of an oblique-angled triangle is 45°, and an What is the smallest value which the adjacent side is √√2. opposite side can have? side is. Solve the triangle when the opposite 5. A ship leaves Cape Cod (42° 2' N., 70° 4′ W.) and sails 200 knots on a course S. 40° E. Find the latitude and longitude reached. *6. If 2 tan 2a = tan 26 sin 2b, find the relation between the tangents of a and b. IX. (Harvard, June, 1893.) (Take the problems in any order. One of the starred problems may be omitted.) 1. What is the base of the system of logarithms, when log 3 0.3976? *2. Solve the right-angled triangle in which one angle is 30°, and the difference of the legs is 4. *3. Find x, given secx= 2 tan x+2. *4. One angle of a triangle is double another angle. The side opposite the first angle is three-halves of the side opposite the second angle. Find the angles. 5. Find, by Middle Latitude sailing, the course and distance from Funchal (32° 38′ N., 16° 54′ W.) to Gibraltar (36° 7' N., 5° 21′ W.). *6. Reduce to its simplest form cos 2 x tan (45° + x)—sin 2x. |