. A treatise on surveying and navigation: uniting the theoretical, the practical, and the educational features of these subjects. PLANE TRIGONOMETRY. 53 PROPOSITION 4. In any triangle, the sines of the angles are to one another as thesides opposite to them. Let ABC be any tri-angle. From the pointsA and B, as centers, withany radius, describe thearcs measuring these an-gles, and drawjpa, CD,and mn, perpendicular to AB. Then, . pa—, mn= By the similar As, Apa and A CD, we have, R : sin.^=5 : CD; or, R(CD)=b (1) By the similar As Bmn and BCD, we have, R : : CD; or, R(CD)=as
. A treatise on surveying and navigation: uniting the theoretical, the practical, and the educational features of these subjects. PLANE TRIGONOMETRY. 53 PROPOSITION 4. In any triangle, the sines of the angles are to one another as thesides opposite to them. Let ABC be any tri-angle. From the pointsA and B, as centers, withany radius, describe thearcs measuring these an-gles, and drawjpa, CD,and mn, perpendicular to AB. Then, . pa—, mn= By the similar As, Apa and A CD, we have, R : sin.^=5 : CD; or, R(CD)=b (1) By the similar As Bmn and BCD, we have, R : : CD; or, R(CD)= (2)By equating the second members of equations (1) and (2).b & Hence, . sin.^4 : : b \ O F D Or, . a : 5=sin A : sin. B) Scholium 1. When either angle is 90°, its sine is radius. Scholium 2. When CB is less than A C, and the angle B, acute,the triangle is represented by A CB. When the angle B becomesB, it is obtuse, and the triangle is A CB; but the proportion isequally true with either triangle; for the angle CBD= CBA,and the sine of CBD is the same as the sine of ABC. In prac-tice we
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Keywords: ., boo, bookcentury1800, booksubjectnavigation, booksubjectsurveying
