M. Cagnoli, in his Trigonometry, has collected into a Table, under one view, and for the purpose of reference, formulæ similar to the preceding. He has also in another Table (which is subjoined) exhibited the various values for the sine, cosine, and tangent of the angle A. 14. 2 2 1- tan. (450) 1 + tan.2 (450 - ) 6. 7. 8. √(1 + cot. A) cos.-sin.. 2 A 2 1-2 sin.2 2.cos.2 A 19 2 1. 9. (1+cos. 24). 10. 11. 12. 13. 1-t 2 1-tan.2 tan. A 2 A 1 + tan.2 2 tan. (45+)+tan. (450-) 14.2 Cos. (450+) Cos. (450) 15. sin. (60° + A) - cos. (60o - А). 15. cos. (60°+A)- cos. (60°-A). The investigation of some of these expressions has been already given; and, by pursuing its plan, the Student will, without difficulty, be able to accomplish that of the others. But the Student, whose object is utility, will feel averse from their investigation, should he suspect them to be mere Trigonometrical curiosities. Such however is not their character; on the contrary, they, in many instances, materially expedite calculation, and furnish to the general language of analysis convenient forms and modes of expression. It is, in accomplishing this latter purpose, that Trigonometrical formulæ are chiefly useful: they serve to conduct investigation where the object has no concern whatever with the properties of triangles. Yet, the investigation of the properties of triangles was the object for which Trigonometry was originally invented; and, if the Student purposes to limit his enquiries to that object alone, he need not, in quest of the requisite formulæ, advance farther than the present Chapter. He may immediately pass on to the fifth Chapter and apply what he has already learned. If, however, his views should extend farther, and he should wish to be possessed of Trigonometry and its formule as instruments of language, he must pursue his researches, become conversant with expressions merely analytical, and, for a time, defer their application. In order that this latter plan may be adopted, we will, in the next Chapter, continue the deduction of Trigonometrical formulæ. F CHAP. III On the Sines, Cosines, &c. of multiple Arcs. - Powers of the Sine and Cosine of the simple Arc. -- Series of the Cosines of Arcs in Arithmetical Progression.-Vieta's, Waring's, and Cotes's Properties of Curves. De Moivre's Expression for the Sine and Cosine of a multiple Arc by means of imaginary Symbols. PROBLEM 5. It is required to express the sine and cosine of twice an arc, in terms of the sine and cosine of the simple arc. By form [1], p. 27, sin. (A + B) = sin. A.cos. B + cos. A. sin. B. Let B = A; ... sin. (2A)=sin. A. cos. A + cos. A. sin. A = 2 sin. A cos. A; or, by the Rule of p. 19, to rad.r, r sin. 2 A = 2 sin. A cos. A* Again, by the form [2] of p. 27, cos. (A + B) = cos. A.cos. B - sin. A.sin. B. Let A = B; ... cos. 2 A = cos. A sin. A; or, = cos. A -(1-cos. A) = 2. cos. A -1; If we employ a radius = r, then, by the Rule of p. 19, r.cos. 2 A = 2 cos.2 2 r2, or = r2 2.sin. A, (see p. 11.) * This result has been (p. 12.) already obtained, but it is here repeated, as being the first of a series of formulæ deduced on the same principle. |