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other words, one is as much greater than the tangent of 45°, as the other is less. In taking, then, the tangent of an arc greater than 45°, we are to suppose the distance between 45 and the division marked with a given number of degrees, to be added to the whole line, in the same manner as if the line were continued out. In working proportions, extending the dividers back, from a less number to a greater, must be considered the same as carrying them forward in other cases. See Art. 185.

Trigonometrical Proportions on the Scale.

188. In working proportions in trigonometry by the scale; the extent from the first term to the middle term of the same name, will reach from the other middle term to the fourth term. (Art. 185.)

In a trigonometrical proportion, two of the terms are the lengths of sides of the given triangle; and the other two are tabular sines, tangents, &c. The former are to be taken from the line of numbers; the latter, from the lines of logarithmic sines and tangents. If one of the terms is a secant, the calculation cannot be made on the scale, which has commonly no line of secants. It must be kept in mind that radius is equal to the sine of 90°, or to the tangent of 45°. (Art. 95.) Therefore, whenever radius is a term in the proportion, one foot of the dividers must be set on the end of the line of sines or of tangents.

189. The following examples are taken from the proportions which have already been solved by numerical calculation.

Ex. 1. In Case I, of right angled triangles, (Art. 134. ex. 1.)

R 45 sin 32° 20′ : 24

Here the third term is a sine; the first term radius is, therefore, to be considered as the sine of 90°. Then the

extent from 90° to 32° 20′ on the line of sines, will reach from 45 to 24 on the line of numbers. As the dividers are set back from 90° to 32° 20'; they must also be set back from 45. (Art. 185.)

2. In the same case, if the base be made radius, (page 60.) R 38 tan 32° 20' 24

Here, as the third term is a tangent, the first term radius is to be considered the tangent of 45°. Then the extent from 45° to 32° 20′ on the line of tangents, will reach from 38 to 24 on the line of numbers.

3. If the perpendicular be made radius, (page 62.) R 24 tan 57° 40′ 38

The extent from 45° to 57° 40′ on the line of tangents, will reach from 24 to 38 on the line of numbers. For the tangent of 57° 40' on the scale, look for its complement 32° 20. (Art. 187.) In this example, although the dividers. extend back from 45° to 57° 40′; yet, as this is from a less number to a greater, they must extend forward on the line of numbers. (Arts. 185, 187.)

4. In Art. 135,

35 R 26: sin 48°

The extent from 35 to 26 will reach from 90° to 48°.

5. In Art. 136,

R 48 tan 274° : 244

The extent from 45° to 2740, will reach from 48 to 244.

6. In Art. 150, ex. 1.

Sin 74° 30′ 32 :: sin 56° 20′ : 271.

For other examples, see the several cases in Sections III.

and IV.

190. Though the solutions in trigonometry may be ef

fected by the logarithmic scale, or by geometrical construction, as well as by arithmetical computation; yet the latter method is by far the most accurate. The first is valuable principally for the expedition with which the calculations are made by it. The second is of use, in presenting the form of the triangle to the eye. But the accuracy which attends arithmetical operations, is not to be expected, in taking lines from a scale with a pair of dividers.*

SECTION VII.

THE FIRST PRINCIPLES OF TRIGONOMETRICAL ANALYSIS.

ART. 191. In the preceding sections, sines, tangents, and secants have been employed in calculating the sides and angles of triangles. But the use of these lines is not confined to this object. Important assistance is derived from them, in conducting many of the investigations in the higher branches of analysis, particularly in physical astronomy. It does not belong to an elementary treatise of trigonometry, to prosecute these inquiries to any considerable extent. But this is the proper place for preparing the formula, the applications of which are to be made elsewhere.

Positive and Negative SIGNS in Trigonometry.

192. Before entering on a particular consideration of the algebraic expressions which are produced by combinations of the several trigonometrical lines, it will be necessary to attend to the positive and negative signs in the different

* See note C.

quarters of the circle. The sines, tangents, &c., in the tables, are calculated for a single quadrant only. But these are made to answer for the whole circle. For they are of the same length in each of the four quadrants. (Art. 90.) Some of them however, are positive; while others are negative. In algebraic processes, this distinction must not be neglected.

193. For the purpose of tracing the changes of the signs, in different parts of the circle, let it be supposed that a straight line CT is

fixed at one end C, while the other end is carried round, like a rod moving on a pivot; so that the point S shall describe the circle ABDH. If the two diameters AD and BH, be perpendicular to each

other, they will di

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194. In the first quadrant AB, the sine, cosine, tangent, &c., are considered all positive. In the second quadrant BD, the sine P'S' continues positive; because it is still on the upper side of the diameter AD, from which it is measured. But the cosine, which is measured from BH, becomes negative, ås soon as it changes from the right to the left of this line. (Alg. 382.) In the third quadrant the sine becomes negative, by changing from the upper side to the under side of DA. The cosine continues negative, being still on the left of BH. In the fourth quadrant, the sine continues negative. But the cosine becomes positive, by passing to the right of BH.

195. The signs of the tangents and secants may be derived from those of the sines and cosines. The relations of these several lines to each other must be such, that a uniform method of calculation may extend through the different quadrants.

In the first quadrant, (Art. 93. Propor. 1.)

R cos tan sin, that is, Tan-

RX sin

COS

The sign of the quotient is determined from the signs of the divisor and dividend. (Alg. 100.) The radius is considered as always positive. If then the sine and cosine be both positive or both negative, the tangent will be positive. But if one of these be positive, while the other is negative, the tangent will be negative.

Now by the preceding article,

In the 2d quadrant, the sine is positive, and the cosine negative.

The tangent must therefore be negative.

In the 3d quadrant, the sine and cosine are both negative. The tangent must therefore be positive.

In the 4th quadrant, the sine is negative, and the cosine positive.

The tangent must therefore be negative.

196. By the 9th, 3d, and 6th proportions in Art. 93.

1. Tan R R cot, that is Cot

:

R2

tan

Therefore, as radius is uniformly positive, the cotangent must have the same sign as the tangent.

2. Cos R R sec, that is, Sec

:

R2

COS

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