. Geometry : the elements of Euclid and Legendre simplified and arranged to exclude from geomtrical reasoning the reductio ad absurdum : with the elements of plane and spherical trigonometry, and exercises in elementary geometry and trigonometry / /c By Lawrence S. Benson. etc., as the angle ACF; and FG is the sine of thisangle; FI, or its equal CG, the cosine; AG the versed sine,and DI the coversed sine; AH the tangent, and CH the secant jDK the cotangent, and CK the cosecant. From these definitions we derive the following corollaries: Cor. 1. The sine, of an angle ACF is half the chord of do
. Geometry : the elements of Euclid and Legendre simplified and arranged to exclude from geomtrical reasoning the reductio ad absurdum : with the elements of plane and spherical trigonometry, and exercises in elementary geometry and trigonometry / /c By Lawrence S. Benson. etc., as the angle ACF; and FG is the sine of thisangle; FI, or its equal CG, the cosine; AG the versed sine,and DI the coversed sine; AH the tangent, and CH the secant jDK the cotangent, and CK the cosecant. From these definitions we derive the following corollaries: Cor. 1. The sine, of an angle ACF is half the chord of doublethe arc measuring it. For if FG be produced to meet the cir-cumference in N, FN is bisected (III. 2) in G, and (III. 17) thearc FAN in A. Cor. 2. The sine of the right angle ACD is the radius CD. Cor. 3. If AF bo half of AD, and consequently ACF half aright angle, the tangent AH is equal to the radius. For A be-ing a right angle, II must be half a right angle, and (I. 1, ) AH equal to AC. Cor. 4. Put the angle = A, and the radius=:l. Then (I. 24,cor. 1) FG-+CG=CF^; that is, siu=A+cosA = l. In like PLANE TRIGONOMETRY. 181 manner, we find from the right-angled triangles CAH, CDK,that Cir=CA^+AH^ and CK^ = CD+DK^; that is, sec^A=l-ftan^A, and cosec^A^l+cot^ Cor. 5. In the similar triangles CGF, CAH, CG : CF, orCA : : CA : CH ; that is, the cosine of an angle is to the ra-dius as the radius to its secant. Hence also (V. 9, cor.) :CA^; that is, cosAsecA = l. It would be found in likemanner from the triangles GIF, CDK, that sin A cosecA=l,CI being equal to the sine FG. Cor. 6. In the same triangles CGF, CAH, the cosine CG isto the sine GF as the radius CA to the tangent AH; whence(V. 8) cos A tanA=sin A. The triangles CIF, CDK give inlike manner sin A cot A = cos A. Cor. 7. The radius is a mean proportional between the tan-gent of an angle and its cotangent. For the triangles CAH,CDK are similar; and therefore HA : AC : : CD, or CA :DK. Hence (V. 9, cor.) tan A cot
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