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If, again, equa. (14) be divided by equa. (13), there

will result,
- #(a + b + c) — bl. [{(a + b + c) — c1

tan AA = Vo #############..(15)

40. Of the rules for the determination of the angles of a plane triangle when the three sides are given, comprehended in the formulae, marked 11, 12, 13, 14, 15, the last three are the best in practice, except the sides are integers and lie within the compass of a table of squares. When the angle sought is small, it is usually better to employ the method of equa. (14) than that of equa. (13). The method of equa. (15) is tolerably commodious, and very correct, except when A is either very small or near 180°. n some cases where great accuracy may be required, the student may wish to obviate the uncertainties that would arise from the use of some of these rules. For this purpose Dr. Maskelyne has given, in the introduction to Taylor's Logarithms, the following rules in reference to the sines and tangents of very small arcs. 1. To find the sine. To the logarithm of the arc reduced into seconds, with the decimal annexed, add the constant quantity 4'6855749, and from the sum subtraet one third of the arithmetical complement of the log. cosine, the remainder will be the log. sine of the given arc, 2. To find the tangent. To the log. arc and above constant quantity, add two-thirds of the arithmetical complement of the log. cosine, the sum is the log, tangent of the given arc. 3. To find the arc, from the sine. To the given log. sine of a small arc add 53144251, and # of the arith. comp. of log. cosine: substract 10 from the index of the sum, the remainder will be the logarithm of the number of seconds and decimals in the given arc. 4. To find the arc from the tangent. To the log. tangent add 53144251, and from the sum subtract 3 of the arith. comp. of log. cosine; take 10 from the index, and there will remain the logarithm of the number of seconds and decimals in the given arc. 41. Having at the end of the 2d chapter adverted briefly to the method of constructing a table of natural

sines and tangents, the subject need not be resumed

here. We shall merely subjoin the values of a few of the sines, cosines, &c in surd expressions, and a few formulae of verification, such as may readily be reduced to identical equations.

‘Now, to radius 1, we have


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- sin
For, since tan = ... we have, for example,

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l l cos 60° = o = $2. :* These may obviously be extended to other-ares, by means of formulae (P), (Q), (R), of this chapter, and prop. 3 of chap. ii. 42. Operations of this kind ought not to be carried . far without being subjected to checks and proofs. For this purpose, after the sines and cosines are found, the -tangents, secants, &c. are easily verified by their mutual trelations. The sines and cosines themselves, are examined by means of some of the following “formulae of overifieation.”

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43. Of the formule investigated in this chapter, those which have letters of reference (A), (3), (c),8c. relate ‘to the sums, differences, multiples, &c. of sines, tan

gents, &c. while those which have figures of reference, (1), (2), (3), &c. will be employed in the solutions of plane triangles. Other kinds will find their application in subsequent parts of this introduction; and the student will do well, after he has gone through their investigation, to arrange them in separate tables for use. e shall terminate the present chapter by subjoining three examples, Example I. Given the three sides of a plane triangle 40, 34, and 25, respectively, to find the largest angle, by formula (15). Here # (a + b + c) c = 15.5... log = 1-1903317 # (a + i + cj – - 1889166i

Their prod. shown by the sum of the logs .. 2.5794978

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Their prod. shown by the sum of the logs... 2.6723288

The latter sum taken from the former

borrowing 20 for radius squared, gives: ****

The latter log. -- by 2 for the v, gives 9,953.5845.

This, by the tables, is nearly tan 41° 563, and by the formula it is tan $A; therefore 3A = 41°56}, and the required angle = 83°53' nearly.

Note. In chap. iii, case 3, the same result is obtained

by a different process.

Example II.

What are thé angles of that plane triangle whose natural tangents are integers? - .

It is evident from equa. (4) that the sum of the three tangents must be equal to their continual product. Now,

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