54. cos e cosh + cos2 0 cosh 20+ cos3 0 cosh 34 + Ρ 9 cos a sinh 28 59. 1 + + 9 12 52 72 112 132 The sum of the reciprocals of the squares of all integers except the multiples of r is (2-1) π2/6r2. 60. The sum of the reciprocals of the squares of all odd integers except the multiples of 2r+ 1 is r(r+1)2 61. The sum of the reciprocals of the squares of the products of all pairs of integers is; and of all pairs of odd integers 68. Hence, using the identity, cosec = cot 10 + 1⁄2 tan 10, 70. Prove that sin 20 = 2 sin 0 cos 0 from the factor expressions of sin ◊ and cos 0. 71. Resolve verse into factors without making use of expression for sin §. 82. Find the general form of the component in 80 and 81. 84. Calculate to 3, 4, 5, 6, 7 decimal places respectively from the formulæ 1, 2, 3, 4, 5 of Art. 539. 1 70 99. We will use the formula = 4 tan-1-tan-1+tan-1 The multiplier, which occurs in 4 tan-1, may be written 4.10-2. 1 Any pair of digits in any power of may be found by adding the preceding pair of digits to the corresponding pair in the preceding power. See above. Then, by Art. 485, sin lies between 0 and 0 - 103 for any acute angle. But if measures 10′′, 0 < .00005, i.e. <1⁄2.10−4, Hence writing 0 for sin 0, the error in sin 10" will be less than. 10-13. That is, For 13 places of decimals, sin 10" = circular measure of 10" 0000484813681. = 552. Similarly we have sin 5"=00002424 nearly by halving the circular measure of 10". Thus sin2 5′′ (2424 × 10−8)2 = 5876 × 10-13, 553. of 10". = (1-cos 10") = '0000000005876, . 2 (1 cos 10") = '0000000023504. To calculate the sines of angles which are multiples The advantage of the above mode of working is that the labour is reduced to the mere multiplication by the small quantity 2- 2 cos a. |