fixed line is equal to its distance from a fixed point? Find the equation. 8. Construct a hyperbola whose transverse axis is 6 and less focal distance 2. Find also the conjugate axis, and the foci and directices of the conjugate hyperbola. LOGARITHMS AND TRIGONOMETRY. I. 3 1. Find the value of the following fraction by logarithms: 0.010006 (1.4 x 40.325062 2. Find the value of the following fraction by logarithms: (0.050395) ) (3.2 % 70.546781 ). using arithmetical complements : C09:00 2 (0.50396) 3. Find the value of the following fraction by logarithms, x : 0.00101904 X 0.99992): 760 X 0.0275142) 4. Define a logarithm. 5. Find, by logarithms, the value of the following quantities to six significant figures : 90.0117283 ; (0.50396)2; No.0117283 ; 2.4 x (0.50396)2: use arithmetical complements in dividing 6. Solve the equation 32+ = 8 by logarithms. 7. Prove that the sum of the logarithms of several numbers is equal to the logarithm of their product. 8. Find, by logarithms, the values of the following quantities to six significant figures: (0.62394); (0.00102173)2; (0.0012173) 0.62394 3.1 x 8(0.62394) ♡ 0.62394 II. 1. In a system of which the base is 9, what is the logarithm of 81 ? of 3 ? of 27 ? of 9 ? of 1 ? of } ? of 31 ? of 0 ? 2. Find, by common logarithms, the values of the following quantities (to five significant figures): 8(0.492162); 1 N0.492162) (0.011009)"; (0.011009369.8 X (0.011009)** 5 Use arith )5 metical complements in dividing. 3. Solve the equation 2048* = 16, by logarithms. 4. Express in a decimal form the numbers which have the following logarithms in a system of which the base is 16:2; -2; – 0.25; 2.75; 0. 5. Find, by common logarithms, the values of the following quantities (to five significant figures): 80.485463); 1 (0.485463) (0.00130106)2; ; Use (0.00130106)2' 2.7 X (0.00130106) arithmetical complements in dividing. 6. Prove that the logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers. 7. Find, by logarithms, the values of the following quantities (to six significant figures): 8(0.0126534); 90.0126534) Use arithmetical complements in divid.204 X (0.56036) ing 8. Solve the equation 243* = 81 by logarithms. 9. What is the characteristic of a logarithm ? 10. What is the logarithm of 1.? of .1? of 1000. ? of .00001 ? of one hundred billionth ? 11. Find, by logarithms, the value of the following ) . V10.0357635) V2.04 + V1.2036) 12. Solve the equation 1024* = 64. 1 (0.56036) quantities (to six significant figures): (0126532) . III. 1. Prove that the logarithm of a quotient is equal to the logarithm of the dividend diminished by the logarithm of the divisor. 2. Find, by logarithms, the values of the following quantities : Ņ0.03478, (0.278)? X (0.008); 0.03478 X (0.7) 3. Prove the formula (sin A)2 + (cos A)2 = 1. What is the formula for the cosine of the sum of two angles? 4. Solve the oblique triangle in which a=50, A = 45°, B= 60° N. B. — a, b, c denote the sides; A, B, C the angles respectively opposite to a, b, c. 5. In a system of logarithms, of which 4 is the base, determine the logarithms of the following numbers : 4; 16; 2; 8; 32; 1; 1= 0.5; 4 = 0.25; }= 0.125; 16 0 1 0.0625; 0. What is the base of the common system of logarithms ? 6. Find, by logarithms, using arithmetical complements, 1 the values of the expressions: (0.001109)2; (0.001109)a i 110.492) X 560 9 X (0.001109)2 7. What single function of any angle A= ? What tan A function is the reciprocal of the secant. 8. Give the formulas for the sine and cosine of the sum and of the difference of two angles; and deduce from these the formulas for the sine and cosine of the double of an angle and of the half of an angle. 9. What is the sine and cosine of 0°, 90°, 180°, 270°, 360°. Work out the formulas for the trigonometric functions of (270° — N). sec A 10. Solve the triangle in which b=0007625, c= 0.015, B= 29° Find both solutions. N.B. - A, B, C denote . the angles respectively opposite the sides a, b, c. IV. 1. What is the logarithm of 1 in any system ? of any number in a system of which that number is the base ? In a system of which the base is 4, what is the logarithm of 64 ? of 2 ? of 8 ? of j ? 2. Find by logarithms, using arithmetical complements, (2 ) the value of the fraction_(0.02183) X (7): V(0.0046) X 23.309 3. Prove the formula for the cosine of the sum of two angles; and deduce the formulas for the cosine of the double of an angle and the cosine of the half of an angle. 4. In what quadrants is the cosine positive, and in what quadrant is it negative ? Prove the values of the cosine of 0°, 90°, 180°, 270°. 5. Given in an oblique triangle b = 0.254, c= 0.317, B= 46°. Solve completely. V. 1. Prove that the logarithm of the product of several factors is equal to the sum of the logarithms of the factors. 2. Prove that the logarithm of the nth root of a number is ath of the logarithm of the number. 3. Work the following examples : 0.01706 x 8.7634 1 x 0.001 ?; ; ?; V4.9 = ?; N0.29 ?; 0.01706 8.7634) X 100 ? Use arithmetical complements in 9 X 10.1109 X (4.9) working the last. |