TRIGONOMETRICAL TABLES. 4. It is to be remembered that the trigonometrical magnitudes defined in the first section as lines are, in reality, numbers; this follows from the supposition that the radius of the circle is equal to the linear unit. The following particular cases serve to place this in a clear point of view: Let it be required to find the values of the sine, cosine, tangent, &c. of the angle 45°, which is equal to half a right angle; by equation (1), But as the triangle BCP is in this case (fig. 5) a right-angled isosceles triangle, it is evident that sin 45° cos 45°, and therefore, = and therefore, I = 2 sin2 45°, I = 2 cos2 45°; sin2 45° = cos2 45° = · If the square root of or 0.5 be extracted to five places of decimals, we find that sin 45° = cos 45° = 0.70710. As the sine and cosine are equal, it follows from equations (4) and (5) that tan 45° = cot 45° = 1. It also appears from (2) and (3), and the values just found for the tangent and cotangent, that and therefore, extracting √2 to five places of decisec 45° = cosec 45° = 1.41421. mals, Let it be required to find the values of the sine, cosine, &c., of 60°. This angle is two-thirds of 90°, or of one right angle, from which it follows that the triangle ACB (fig. 6) is equilateral, and therefore that the radius AC is bisected in P; but as the radius = 1, CP = 1; and therefore From this value of the cosine we obtain that of the sine, as in example 1, p. 9: sin 60° = √√ (1 − 1) = √ † ; extracting the square root of 2 = 0.75, to five places, Let it be required to find the sine and cosine of 30°. As 30° is the complement of 60°, it follows from section 3, that These cases are sufficient to show, that trigonometrical magnitudes are numbers, which are capable of being calculated from geometrical principles, and accordingly, tables, called Tables of Natural Sines, have been computed with great accuracy, in which the values of the sines, cosines, tangents, secants, &c., of every degree and minute in the quadrant are registered; by means of which, and the application of a few easy rules, the value of the sine, cosine, &c., of any given angle, may be found. The statement of the method by which such tables are constructed is unsuited to the present treatise. The mode of using them in computations is explained in the Appendix. 4. Calculate the sine and cosine of 120°. As 120 is the supplement of 60°, it follows from sect. 3, that Ans. sin 120° = sin 60° = 0.86602, cos 120°-cos 60°=-0.50000. 5. Calculate the tangent of 120°. If ACB and ACB' (fig. 7) be each 18°, BCB' is equal 36°, and as 36' is contained ten times in 360°, it follows that BB' is the side of a regular decagon inscribed in the circle, and therefore that it is equal to the greater segment of the radius (= 1), cut in extreme and mean ratio (Euclid, Book IV. Prop. x.); but the greater part of unity cut 9. Calculate the sine, cosine, and tangent of 72°. Since 72° is the complement of 18°, Ans. sin 72° 0.95105 cos 72° 0.30901 tan 72°3.07768. CHAPTER III. RIGHT-ANGLED TRIANGLES. 1. Relations between the Sides and Angles.-2. Four Cases of RightAngled Triangles.-3. The Four Cases computed by Logarithmic Tables. RELATIONS BETWEEN THE SIDES AND ANGLES. 1. LET ABC (fig. 8) be a right-angled triangle, and let AN be measured off upon the hypotenuse, equal in length to the linear unit. With A as centre, and AN as radius, describe an arc of a circle, NM; draw NP and MT perpendicular to AC, then NP is the sine and MT the tangent of the angle A. Since the triangles BAC and NAP are similar, BC: AB:: NP: AN, and therefore :: sin A : 1; BC AB × sin A. = As the angle B is the complement of A, it follows (Chap. II. sect. 3) that Similar values may be obtained for the side AC, and therefore PROPOSITION I. In a right-angled triangle, either side is equal to the hypotenuse, multiplied by the sine of the opposite, or cosine of adjacent, angle. |