an important proposition, which enunciated in words is, The sine of an angle is cqual to the sine of its supplement. that is, The cosine of an angle and the cosine of its supplement are equal in absolute magnitude, but have opposite signs. If, as in the annexed figure, we draw CP', making with Ca an angle aCP' equal to the angle, we shall find, in like manner, sin. (180°) = .sin. P If we draw CP', making with Cỏ an angle ¿CP' = 4, then as is evident from the Def. 7, and the rule for signs; and, in like manner, we may proceed for angles in the fourth quadrant. These relations being established between the sines and cosines, the corresponding relations between the other trigonometrical functions, may be deduced immediately from Table I. Thus, tan. (90° + 1) = sin. (90°) cos. oot./ P The student may exercise himself, by verifying such of the results in the following table as have not been formally demonstrated. The results in the above table which are most frequently used, are marked with an asterisk, and ought to be committed to memory. We have in the preceding pages confined ourselves to the consideration of angles not greater than 360°, but the student can find no difficulty in applying the above principles to angles of any magnitude whatsoever. We shall conclude this introductory chapter, by demonstrating two propositions which are of the highest importance in our subsequent investigations. 'The first is, In any right-angled triangle, the ratio which the side opposite to one of the acute angles bears to the hypotenuse, is the sine of that angle; the ratio which the side adjacent to one of the acute angles bears to the hypotenuse, is the cosine of that angle; and the ratio which the side opposite to one of the acute angles bears to the side adjacent to that angle, is the tangent of that angle. Let CMP be any plane triangle, right-angled P In any plane triangle, the ratio of any two of the sides, is equal to the ratio of the sines of the angles opposite to them. Let ABC be a plane triangle; it is required to prove, that CB sin. A CB = = sin. A CA sin. B = CA sin. B' BA sin. C' BA sin. C' From C let fall CD perpendicular on AB. Then, since CDB is a plane triangle right-angled at D, by last proposition, CD = CB sin. B CD = CA sin. A Again, since CDA is a plane triangle right-angled at A, In like manner, by dropping perpendiculars from B and A upon the sides AC, CB, we can prove, CB In treating of plane triangles, it is convenient to designate the three angles by the capital letters A, B, C, and the sides opposite to these angles by the corresponding small letters, a, b, c. According to this notation, the last propo sition will be, Given the sines and cosines of two angles, to find the sine of their sum. Let ABC be a plane triangle; from C let fall CD p per pendicular on AB, But, since ABC is a plane triangle, ++ C = 180° = sin. {180° — (0 +3 (0 + 0). sin (+), because 180°-(+) is the supplement of Hence, sin. (+0) = sin. cos. sin. cos. .......................................... (a) This expression, from its great importance, is called the fundamental formula of Plane Trigonometry, and nearly the whole science may be derived from it. fhren the sines and cosines of two angles, to find the sine of their difference. By formula (a). sin, (+) = sin. cos. - sin. cos. & For substitute 180° 6, the above will become Substitute, therefore, these values in the above expression, it becomes sin. (0 0) = sin. cos. - sin. cos. é ...... (b) Given the sines and cosines of two angles, to find the cosine of their sum. By formula (a) sin. (0+ 0) = sin. cos. + sin. cos. 6. For substitute 90° +6, the above will become cos. (0 +0') = cos. 8. = sin. 8. Substituting, therefore, these values in the above expression, it becomes Given the sines and cosines of two angles, to find the cosine of their difference. Substituting, therefore, these values in the above expression, it becomes cos. (—) = cos. cos. + sin. sin. .................. Given the tangents of two angles, to find the tangent of their sum. (c) (d) |