54.

- sinh 2a + sinh 3a - ...

cos e cosh + cos2 0 cosh 20+ cos3 0 cosh 34 +

59.

1 1 1

12 52 72 112

The sum of the reciprocals of the squares of all integers except the multiples of r is (2-1) 2/62.

60. The sum of the reciprocals of the squares of all odd r(r+1)2

integers except the multiples of 2r+ 1 is

2 (2r+ 1)"

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,

71. Resolve vers into factors without making use of expression for sin 0.

40

40

40

1+

5 п

73. sin 0 + cos 0 = (1 + 10) (1-10) (1 + 10) (1-40)...

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.

85. Calculate

of Art. 533.

86. The equation this root lies between 1

to 5 decimal places from the formulæ

= cos has one and only one root: and and π.

The multiplier, which occurs in 4 tan-1, may be written. 4.10-2.

1 99

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.

Now let = circular measure of 10′′ ÷ 90 ÷ 60 ÷ 6 = above calculated decimal.

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,

03:10-12.10-13.

Hence writing for sin 0, the error in sin 10" will be less than 1.10-13. 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".

-13

Thus sin2 5′′ = (2424 × 10−8)2 = 5876 × 10−13,

..(1-cos 10")=0000000005876,

553. To calculate the sines of angles which are multiples

of 10".

If a denotes any angle, we have

sin (n + 1) a + sin (n − 1) a = 2 sin na cos a,

sin na sin (n − 1) a − (2 – 2 cos a) sin na.

.'. sin (n + 1) a − sin na = In this equation let a =

2-2 cos a.

10". Then we have found sin a and

Putting n = 1 gives us sin 20′′ – sin 10".
Putting n 2 gives us sin 30′′ – sin 20′′.

And so on.

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.