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RIGHT ANGLED PLANE TRIANGLES.
12. 1. Right angled triangles may, as well as others, be solved by means of the rule to the respective case under which any specified example falls: and it will then be found, since a right angle is always one of the data, that the rule usually becomes simplified in its application; as appeared in the solution of the second example to case 2. 2. When two of the sides are given, the third may be found by means of the property demonstrated in Euc. i. 47. Thus, Hypoth. = y (base’ + perp.”.) Base = V (hyp.” – perp.”) = M(hyp. + perp.). (hyp. – perp.) Perp. = 1/(hyp.” – base”) = A/(hyp. Thase). (Byp. F base). 3. There is another method for right angled triangles, known by the phrase making any side radius; which is this. “To find a side, Call any one of the sides radius, and write upon it the word radius; observe whether the other sides become sines, tangents, or secants, and write those words upon them accordingly. Call the word written upon each side the name of each side: then say, “As the name of the given side, “Is to the given side; “So is the name of the required side, “To the required side.” “To find an angle. Call either of the given sides radius, and write upon it the word radius; observe whether the other sides become sines, tangents, or secants, and write those words on them accordingly. Call the word written upon each side the name of that side. Then say, “As the side made radius, “Is to radius; “So is the other given side, “To the name of that side: which determines the opposite angle.”
13. When the numbers which measure the sides of the triangle, are either under 12, or resolvable into factors which are each less than 12, the solution may be obtained, conformably with this rule, easier without logarithms than with them. For, Let ABC be a right angled triangle, in C which AB the base is assumed to be radius; Bc is the tangent of A, and Ac its secant, to that radius; or, dividing each of these by the base, we shall have the tangent and A B secant of A, respectively, to radius 1. Tracing in like manner, the consequences of assuming Bc, and Ac, each for radius, we shall readily obtain these expressions.
14. In a right angled triangle are given, the hypothenuse and the base, 25 and 24 respectively; to find the rest.
1. IN the preceding o the investigation of trigonometrical properties has been conducted geometrically; the various relations of the sines, cosines, tangents, &c. of arcs and angles, whether depending upon triangles or not, being deduced immediately from the figures to which the several enquiries were referred. This method carries conviction at every step; and by keeping the objects of enquiry constantly before the eye of the student, serves admirably to guard him against the admission of error; and is, therefore, well fitted for adoption at the commencement of this branch of science. But of late years, another method, first introduced by Euler, has been generally employed by the continental mathematicians, and very frequently by those of England. It is analytical. The nature and mutual relation of the lineo-angular quantities, sines, tangents, &c. being defined by a few obvious and simple equations, every other theorem and formula that is likely to be of use, is deduced with great facility by the mere reduction and transformation of the original equations. This method serves greatly to shorten almost all trigonometrical investigations, except a few which lie at the foundation of the science; and admitting of an extension to which the geometrical method cannot lay claim, at the same time that it proceeds to some of its most important results with great rapidity; no treatise on trigonometry can be complete that does not assign to this manner of handling the subject a considerable place. 2. Previously to the student's entrance upon this department of enquiry, it will be expedient for him to trace the mutations which the principal intermediary or lineo-angular quantities undergo, when they relate to arcs found in different quadrants of the same circle. To this end let him draw afresh and lay before him, the diagram given at chap. i. art. 18, and first trace the mutations of the sines and the cosines; keeping in mind this general principle, that every variable algebraic quantity changes its sign after it becomes 0. 3. Let the arc be supposed to commence at A, and to increase in the direction ABEA’B. As the arc augments through the first quadrant AE, the sine augments till it becomes equal to CE, the radius; passing from 90° to 180° through the quadrant EA, the sine diminishes, but is to be regarded as positive till the are becomes AEA." or 180°. In that state of the arc the value of the sine is obviously 0. Passing on to the third quadrant, as when the arc is AEA'B', the sine D'B' is directed contrarily to what it was in the first two quadrants, and is then to be regarded as negative. In this state it continues to increase through the third quadrant, at the end E' of which it is again equal to radius. From thence the negative value of the sine diminishes, till at the end A of the fourth arc, it again passes through zero, and the sine becomes positive in the fifth quadrant; as it obviously ought to do, since the fifth quadrant is coincident with the first. In the 1st and 2d quadrants, then, the sines are +, In the 3d and 4th, they are –. If the arc were taken negatively, its sine would, for like reasons, be negative in the first half, positive in the second half, of the circle. 4. The cosine of an arc is equal to radius at the point A, where the arc is evanescent. From thence it dimimishes while the arc increases through the first quadrant; at E the cosine is 0. Then it changes its sign and continues negative through the arc EA'E', that is, from 90° to 270°. At E' the cosine is again 0; and, of course, from E' to A, through the fourth quadrant, it is again positive. The rule is the same for a negative arc. 5. With regard to the signs of the tangents, it is evi
dent, since tan = . (chap. i. 19), that when the signs
of the sine and cosine are both alike, whether positive or negative, the sign of the tangent is positive; and when those signs are unlike, the sign of the tangent is negative. The tangent, therefore, is positive in the first quadrant, the sine and cosine being then both positive: it is negative in the second quadrant, the cosine being then negative though the sine remains positive; it becomes positive again in the third quadrant, the sine and cosine being then both negative: finally, it is negative