X. (Harvard, September, 1893.) (One of the starred problems may be omitted.) 1. If the base of our system of logarithms were 20 instead of 10, what would be the logarithm of one tenth? *2. The area of a right triangle is 6, and the sum of the three sides is 12. Solve the triangle. *3. Reduce to its simplest form cos2 B+sin2 B cos 2 A-sin? A cos 2 B. *4. Two angles of a triangle are sum of the two opposite sides is 10. 40° 14' and 60° 37'. The (34° 22' S., 18° 30' E.), Find, by Middle Lati 5. A ship leaves Cape of Good Hope and sails N. 40° W. to Latitude 30° S. tude Sailing, the Longitude reached and the distance sailed. *6. The base angles of a triangle are 22° 30' and 112° 30'. Find the ratio between the base and the height of the triangle. XI. (Harvard, June, 1894.) (Arrange your work neatly.) 1. What is meant by the logarithm of a number ʼn in the system whose base is 8? What will be the logarithm of 4 in this system? Which sign should be used when a lies in the first quadrant? When a lies in the second quadrant? 3. In a triangle two angles are equal to 32° 47′ and 49° 28' respectively and the length of the included side is 0.072. Solve the triangle. 4. A circular tent 30 feet in diameter subtends at a certain point an angle of 15°. Find the distance of this point from the centre of the tent. 5. A ship leaves Latitude 42° 2' N., Longitude 70° 3′ W., and sails N. 40° E. a distance of 420 miles. Find by Middle Latitude Sailing the position reached. XII. (Sheffield Scientific School, September, 1892.) 1. Express an angle of 60° in radians. 2. Represent geometrically the different trigonometric functions of an angle. State the signs of each function for each quadrant. 3. Express tan and sec in terms of sin p. 4. Derive the formula sin asin ẞ2 sin (a + B) cos (a-B). 5. Show that, if a, b and c are the sides of a triangle and A is the angle opposite the side a, then a2=b2+c2-2 be cos A. bc 6. Given cos 2 x2 sin x, to find the value of sin x. 7. Given two sides of a triangle a=450.2, b=425.4, and the included angle C=62° 8'; find the remaining parts. XIII. (Sheffield Scientific School, June, 1893.) 1. Express an angle of 15° in radians. 2. Write the simplest equivalents for sin (+), tan (27—д), cos (3π-4), sec (π+6). 3. Express tan in terms of sin o, cos o and cot o, respectively; and cos in terms of tano, seco and coseco, respectively. 4. Show (a) that sin (a+ẞ) + sin (a—ß) = 2 sin a cos ß; (b) that cos (a+ß)+cos (a− ß) = 2 cos a cos ß. 6. Obtain a formula for tana in terms of cos a. 7. The base of a triangle c=556.7, and the two adjacent angles a=65° 20′.2, ß=70° 00'.5; calculate the area of the triangle. 8. Given 0<a<90°, and log cos a=1.85254, to determine a. XIV. (Sheffield Scientific School, September, 1893.) 1. Reduce an angle of 3.5 radians to degrees. 2. Define the different trigonometrical functions of an angle and give their algebraic signs for an angle in each quadrant. 3. Write simple equivalents for the following functions: sin(-a); cos (-a); tan (+α); sec (—α). 4. Express cosec a in terms, respectively, of sin a, cos a, tan α, cot a, sec a. 5. Reduce (cos a cos ẞ— sin a sin ß)2+ (sin a cos ẞ+cos a sin ß)2 to its simplest equivalent. 7. The sum of two sides, a and b, of a triangle is 546.7 ft., the sum of the opposite angles, a and ẞ, is 124°, and sin a: sin ß =1.003; find the angles and sides of the triangle. 8. Given 0<a<90°, and log cot a=0.03293, to determine a. XV. (Sheffield Scientific School, June, 1894.) 1. Express (a) an angle of 2 radians in degrees; (b) an angle of 30° in radians. 2. Give simple equivalents for the following functions: tan (—x), cosec (−x), sin(x+1π), sin(x — 1⁄2π), tan(3π—x), sin (2π-x). 6. Given 180° <$<270°, and log cot p=0.3232, find 4. 7. The sides of a triangle are a=32.5 ft., b=33.1 ft., c=32.4 ft.: Calculate the area of the triangle and the angle C opposite the side c, using the following formula: S =√p (p − a) (p − b ) ( p − c) = ±ab sin C, in which S denotes the area of the triangle, and p=(a+b+c). CHAPTER VI. CONSTRUCTION OF TABLES. § 42. LOGARITHMS. Properties of Logarithms. Any positive number being selected as a base, the logarithm of any other positive number is the exponent of the power to which the base must be raised to produce the given number. an Thus, if a2 = N, then n = logɑN. This is read, n is equal to log N to the base ɑ. Let a be the base, M and N any positive numbers, m and n their logarithms to the base a; so that 3. The logarithm of the reciprocal of a positive number is the negative of the logarithm of the number. |