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94. Other relations of the sine, tangent, &c. may be derived from the proposition, that the square of the hypothenuse is equal to the sum of the squares of the perpendicular sides. (Euc. 47. 1.)
In the right angled triangles CBG, CAD, and CHF, (Fig. 3.)
1. CGʻ=CB' +BG”, that is, R?=cos? +sin?,*
The sine of 90
The chord of 60° y are, in any circle, each equal to And the tangent of 45° the radius, and therefore equal to each other.
1. In the quadrant ACH, (Fig. 5.) the arc AH is 90°. The sine of this, according to the definition, (Art. 82.) is CH, the radius of the circle.
2. Let AS be an arc of 60°. Then the angle ACS, being measured by this arc, will also contain 60°; (Art. 75.) and the triangle ACS will be equilateral. For the sum of the three angles is equal to 180°. (Art. 76.) From this, taking the angle ACS, which is 60°, the sum of the remaining two is 120°. But these two are equal, because they are subtended by the equal sides, CA and CS, both radii of the circle. Each, therefore, is equal to half 120°, that is to 60°.
* Sine is here put for the square of the sine, cos' for the square of the cosine, &c.
All the angles being equal, the sides are equal, and therefore AS, the chord of 60°, is equal to CS the radius.
3. Let AR be an arc of 45°. AD will be its tangent, and the angle ACD subtended by the arc, will contain 45o. The angle CAD is a right angle, because the tangent is, by definition, perpendicular to the radius AC. (Art. 84.) Subtracting ACD, which is 45°, from 90°, (Art. 77.) the other acute angle ADC will be 45° also. Therefore the two legs of the triangle ACD are equal, because they are subtended by equal angles ; (Euc. 6. 1.) that is, AD the tangent of 45°, is equal to AC the radius.
Cor. The cotangent of 45° is also equal to radius. For the complement of 45° is itself 45°. Thus HD, the cotangent of ACD, (Fig. 5.) is equal to AC the radius.
96. The sine of 30° is equal to half radius. For the sine of 30° is equal to half the chord of 60°. (Art. 82. cor.) But by the preceding article, the chord of 60° is equal to radius. Its half, therefore, which is the sine of 30° is equal to half radius.
Cor. 1. The cosine of 60° is equal to half radius. For the cosine of 60° is the sine of 30°. (Art. 89.)
Cor. 2. The cosine of 30°=;V3. For
Cos? 30°=R? - sina 30o=1-=
1 Therefore, Sin 45o=V1=
97. The chord of any arc is a mean proportional, between the diameter of the circle, and the versed sine of the arc.
Let ADB (Fig. 6.) be an arc, of which AB is the chord, BF the sine, and AF the versed sine. The angle ABH is a right angle, (Euc. 31. 3.) and the triangles ABH and ABF are similar. (Euc. 8. 6.) Therefore,
AH : AB :: AB: AF.
That is, the diameter is to the chord, as the chord to the versed sine.
In Fig. 6th, let the arc AD=a, and ADB=2a. Draw BF perpendicular to AH. This will divide the right angled triangle ABH into two similar triangles. (Euc. 8. 6.) The angles ACD and AHB are equal, (Euc. 20. 3.) Therefore the four triangles ACG, AHB, FHB, and FAB are similar; and the line BH is twice CG, because BH : CG:HA: CA.
The sides of the four triangles are,
HF=vers. sup. 2a,
A variety of proportions may be stated, between the ho mologous sides of these triangles : For instance,
By comparing the triangles ACG and ABF, AC: AG::AB : AF, that is, R : sin a:: 2sin a : vers 2a AC: CG::AB : BF
R: cos a:: 2sin a ; sin 2a AG: CG::AF: BF Sin a : cos a::vers 2a : sin 2a
R X sin 2a =2sin a Xcos a
By comparing the triangles ACG and BFH, AC: CG::BH: HF, that is, R : cos a:: 2cos a : vers.sup. 2a AG: CG::BF : HF Sin a : cos a::sin 2a : vers. sup. 2a
That is, the product of radius into the versed sine of the supplement of twice a given arc, is equal to twice the square of the cosine of the arc.
And the product of the sine of an arc, into the versed sine of the supplement of twice the arc, is equal to the product of the cosine of the arc, into the sine of twice the arc, &c. &c.
THE TRIGONOMETRICAL TABLES.
Art. 98. To facilitate the operations in trigonometry, the sine, tangent, secant, &c. have been calculated for every degree and minute, and in some instances, for every second, of a quadrant, and arranged in tables. These constitute what is called the Trigonometrical Canon.* It is not necessary to extend these tables beyond 90°; because the sines, tangents, and secants, are of the same magnitude, in one of the quadrants of a circle, as in the others. Thus the sine of 30° is equal to that of 150°. (Art. 90.).
99. And in any instance, if we have occasion for the sine, tangent, or secant of an obtuse angle, we may obtain it, by looking for its equal, the sine, tangent, or secant of the supplementary acute angle.
100. The tables are calculated for a circle whose radius is supposed to be a unit. In may be an inch, a yard, a mile, or any other denomination of length. But the sines, tangents, &c. must always be understood to be of the same denomination as the radius.
101. All the sines, except that of 90°, are less than the radius, (Art. 32, and Fig. 3.) and are expressed in the tables by decimals.
Thus the sine of 20° is 0.34202, of 60° is 0.86603,
of 40° is 0.64279, of 89° is 0.99985, &c. When the tables are intended to be very exact, the decimal is carried to a greater number of places.
The tangents of all angles less than 45o are also less than radius. (Art. 95.) But the tangents of angles greater than 45°, are greater than radius, and are expressed by a whole number and a decimal. It is evident that all the secants also
* For the construction of the Canon, see Sect
must be greater than radius, as they extend from the center, to a point without the circle.
102. The numbers in the table here spoken of, are called natural sines, tangents, &c. They express the lengths of the several lines which have been defined in arts. 82, 83, &c. By means of them, the angles and sides of triangles may be accurately determined. But the calculations must be made by the tedious processes of multiplication and division. To avoid this inconvenience, another set of tables has been provided, in which are inserted the logarithms of the natural sines, tangents, &c. By the use of these, addition and subtraction are made to perform the office of multiplication and division. On this account, the tables of logarithmic, or as they are sometimes called, artificial sines, tangents, &c. are much more valuable, for practical purpose, than the natural sines, &c. Still it must be remembered, that the former are derived from the latter. The artificial sine of an angle, is the logarithm of the natural sine of that angle. The artificial tangent is the logarithm of the natural tangent, &c.
103. One circumstance, however, is to be attended to, in comparing the two sets of tables. The radius to which the natural sines, &c. are calculated, is unity. (Art. 100.) The secants, and a part of the tangents are, therefore, greater than a unit; while the sines, and another part of the tangents, are less than a unit. When the logarithms of these are taken, some of the indices will be positive, and others negative; (Art. 9.) and the throwing of them together in the same table, if it does not lead to error, will at least be attended with inconvenience. To remedy this, 10 is added to each of the indices. (Art. 12.) They are then all positive. Thus the natural sine of 20° is 0.34202. The logarithm of this is 1.53405. But the index, by the addition of 10, becomes 10–159. The logarithmic sine in the tables is therefore 9.53405.* Directions for taking Sines, Cosines, d-c. from the tables.
104. The cosine, cotangent, and cosecant of an angle, are the sine, tangent, and secant of the complement of the angle. (Art. 89.) As the complement of an angle is the difference between the angle and 90', and as 45 is the half of 90; if any given angle within the quadrant is greater than 45°, its
* Or the tables may be supposed to be calculated to the radius 10000000000, whose logarithm is 10.