An Analytical Treatise on Plane and Spherical Trigonometry, and the Analysis of Angular Sections

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John Taylor, 1828 - 317 pages

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Page 80 - In a right angled spherical triangle, the rectangle under the radius and the sine of the middle part, is equal to the rectangle under the tangents of the adjacent parts ; or, to the rectangle under the cosines of the opposite parts The...
Page 20 - In any plane triangle, the sum of any two sides is to their difference as the tangent of half the sum of the opposite angles is to the tangent of half their difference.
Page 48 - The sum of the three sides of a spherical triangle is less than the circumference of a great circle. Let ABC be any spherical triangle; produce the sides AB, AU, till they meet again in D.
Page 67 - Let a, b, c, be the sides, and A, B, c, the angles of a spherical triangle, as usual.
Page 58 - In any spherical triangle, the greater side is opposite the greater angle ; and conversely, the greater angle is opposite the greater side.
Page 30 - The logarithm of a power of a number is found by multiplying the logarithm of the number by the exponent of the power. For, A« = (10°)
Page 93 - Б) arc of the same species. 180. In the solution of oblique-angled spherical triangles, there are six cases, the data in them being, respectively, I. Two sides and an angle opposite one of them. II. Two angles and a side opposite one of them. III. Two sides and the included angle. IV. Two angles and the included side. V. The three sides. VI. The three angles. CASE I. 181. Given two sides and an angle opposite one of them. Let there be given, in the oblique- c angled spherical triangle ABC, the sides...
Page 81 - ... is enabled to solve every case of right-angled triangles. These are known by the name of Napier's Rules for Circular Parts ; and it has been well observed by the late Professor Woodhouse, that, in the whole compass of mathematical science, there cannot be found rules which more completely attain that which is the proper object of all rules, namely, facility and brevity of computation.
Page 24 - ... and using, in case of need, the auxiliary angles ; with the modifications of the formulae whence they are derived ; and having reference, for convenience of notation, to a spherical triangle ABC, figure 15 or 16, whose sides a, b, c, are respectively opposite to the angles A, B, C.
Page 72 - ... b) = sec b, cosec ( — b) = — cosec b ; (53) that is, the cosine and secant of the negative of an angle are the same as those of the angle itself ; and the sine, tangent, cotangent, and cosecant of the negative of an angle are the negatives of those of the angle. These results correspond with those obtained geometrically (Art. 68). 80.

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