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Thus, in interpolating a value of f(c) when the value of x is given, the degree of accuracy varies inversely with this second term which involves d?.

But, in interpolating a value of x when the value of f(x) is given, the degree of accuracy varies directly with the first term which involves d.

585. When the term involving d' is so large as to inconveniently restrict the degree of accuracy with which a value of f (ə) may be interpolated, then the differences in f(x) are said to be irregular.

When the term involving d is so small as to inconveniently restrict the degree of accuracy with which a value of x may be interpolated, then the differences in f(a) are said to be insensible.

586. In the preceding two articles it is essential to note that the possibility of applying the formula of proportional differences depends primarily upon the smallness of the second term involving d.

Hence the absolute smallness of this term measures the accuracy for interpolating f (x) from x.

And the relative smallness of this term as compared with the first term measures the accuracy for interpolating x from f (x).

587. The general conclusions are summed up by using the terms irregular and insensible.

I. The differences in sin a are insensible as a approaches 90° : hence the accuracy for interpolating a from sin a diminishes as a increases.

II. The differences in tan a and in sec a are irregular as a approaches 90°; hence the accuracy for interpolating tana or sec a from a diminishes as a increases : and the differences in tan a and sec a are insensible as a approaches zero; hence the accuracy for interpolating a from tan a or sec a diminishes as a diminishes.

III. The differences in log sin a and log tan a are irregular as a approaches zero; hence the accuracy for interpolating log sin a or log tan a diminishes as a diminishes : and the differences in log sin a and log tan a are insensible as a approaches 90°; hence the accuracy for interpolating a from log sin a or log tan a diminishes as a increases.

a

Errors in measurement.

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588. The determination of a required length or angle by means of trigonometrical formulæ illustrates the value of the investigations of this chapter.

Thus the required magnitude is a certain function of magnitudes directly measured. But any magnitude directly measured is subject to an error, whose limit is generally known. Hence it becomes important to evaluate the possible error in the derived magnitude.

A few examples will illustrate this point.

Example 1. The height h of a tower is determined by measuring a horizontal line a from its base and the angle of elevation of the top of the tower at the end of the line. Find the possible error in the height h due to the possible error 8 in the angle 0. Here we have hra tan 0 for estimated height

and h'=a tan (@+8) for real height. .. possible error, i.e. h' h=a{tan (0+0) – tan 0}

=a8 sec approximately. Hence the absolute possible error varies as sec20, and therefore increases as O increases. But the relative possible error, i.e. the ratio of the absolute possible

að sec2 A error to the whole height

= 28 cosec 20 : which is least when

a tan A=45°.

Example 2. The height h of a hill is determined by measuring the angles of elevation a and ß of the top and bottom of a tower of height b on the top of the hill. Find the possible error in h due to a possible error d in the angle a.

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a

a

b cota

..h

Here we have (h+b) cot a=h cot B,

6
cot ß-cot a tan a cot ß-1'

b.
and h' =

tan (a+0) cotß-1'
h-h' tan (a+8) cot ß-1

- 1
h tan a cot ß-1
8 secé a cotß

approximately,
tan a cotß-1

=8{cot(a-B)+tan a}, which gives the relative error.

EXAMPLES XX.

1. Show that the equation sin 0 = 0 may be used with the following degrees of accuracy:

From zero to 18 up to 7 places : from zero to 52' up to 6 places; from zero to 95' up to 5 places; from zero to 164' up to 4 places.

Hence calculate the sines of 7' 12"; 30' 15"; lo; and 2° 11' 2" as accurately as is possible from the formula sin 0 = 0.

2. Given that the moon subtends at the earth an angle of half a degree, find the distance at which a circular plate of six inches diameter must be placed so as just to conceal it.

3. Given that the radius of the earth is 3960 miles, show that the number of miles in the visible horizon from a point of observation at a given number of feet above the ground may be found, approximately, by adding to the number of feet its half and taking the square root of the result.

4. Given that the tangents of 32° 11', 32° 12', 57° 48', 57° 49' are respectively •6293274, •6297336, 1.5879731, 1.5889979, find the cosecant of 64° 22' 26".

=

5. Given that sin (45° + A) = 71, show that A is, approximately, 14' or 89° 46', the circular measure of l' being .0002909.

sin A 1733 6. Given that

show that A is the circular mea

0 1734' sure of 3° 22' approximately.

7. Given that the tangents of 21° 20' and 34° 20' are •3905541 and •6830066 respectively, find the tangent of 55° 40'.

8. From a table of tangents calculated for every minute, show that an angle may be determined to within about oth part of a second when the angle is about 30° : and to within about th part of a second when the angle is about 60°.

9. From a table of log sines calculated for every minute, show that an angle may be determined to within about s of a second when the angle is about 88° or about 33°.

10. Find L sin 1° 40' 15" by interpolating in the table of log sines directly. Then find its value by Maskelyne's method : and hence compare the degrees of accuracy of the two methods.

11. Find the angle whose log sine is 2.7523456.

12. If the angle C of a triangle be very obtuse, the number of seconds in its defect from 180° is very nearly

12 (a + b)(a + bc) 206265

ab

a

20

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13. An angle a +8 is found by interpolating in the table of logarithmic sines given for every minute.

Show that the circular measure of the error involved is, approximately,

(ud - 82) cosec 2a, where

M

is the circular measure of 1'. Hence show that the maximum error, in calculating an angle between a and a +1', is about 0":00435 cosec 2a.

a

14. If the area of a triangle is calculated by measuring two sides and the included angle as 197 ft., 2030 ft., and 60° respectively, and if there is an error of 2" in the measurement of the angle, show that this will involve an error of about 1 sq. ft. in the area.

15. If the angles B and C of a triangle are found from A, b, c: and if there is a small error 8 in A; show that the errors in B and C are to one another as tan B to tan C: and hence that they are, respectively,

- 8 sin B cos C cosec A and, -8 sin C cos B cosec A. 16. If errors a', b, c are made in the measurement of the three sides a, b, c of a triangle, show that the error in A is

- a (bcos C + d' cos B-a)/S. 17. If an error a' is made in a and no errors are made in 6 and c; then r', ri', R', the errors involved in r, ru, R respectively, will be given by

r = a' (R/8 žra); ri' = a'(R/8 }r);

R' = á cot B cot C cosec A. 18. The height h of an object is estimated by measuring its angles of elevation a, b, y at three points A, B, C in a straight line, such that BC = a, AC =b, AB=C. If a', B', y be the errors in a, b, y, show that the error in h is h3

b cos ß a'

ß' + abc ( sind

siny

a cos a

C COS Y

.

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sin? ß

a

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