Elements of geometry and trigonometry . 1?D and give the pro- 25S PLANE TRIGONOMETRY. But we also have, CE : CFR : cos C C07^ : CB : CA : hence: CB : CA. If the radius R=l, we shall have, AB = CB sin C, and CA=CB cos C. Hence, in every right angled triangle, the perpendicular is equalto the hypothenuse hyihe siiie of the angle at the hase ;and- the base is eqwd to the hypothenuse multiplied by the cosineof the angle at tJie base ; the radius being equal to unity. THEOREM In every right angled triangle^ radius is to the tangent of ei-ther of the acute angles, as the side adjacen
Elements of geometry and trigonometry . 1?D and give the pro- 25S PLANE TRIGONOMETRY. But we also have, CE : CFR : cos C C07^ : CB : CA : hence: CB : CA. If the radius R=l, we shall have, AB = CB sin C, and CA=CB cos C. Hence, in every right angled triangle, the perpendicular is equalto the hypothenuse hyihe siiie of the angle at the hase ;and- the base is eqwd to the hypothenuse multiplied by the cosineof the angle at tJie base ; the radius being equal to unity. THEOREM In every right angled triangle^ radius is to the tangent of ei-ther of the acute angles, as the side adjacent to the side op-posite. Let CAB be the proposed tri-angle. With any radius, as CD, de-scribe the arc DE, and draw thetangent DG. From the similar trianglesCDG, CAB, we shall have, CD : DG : : CA : AB : hence,R : tanff C : : CA : AB. Cor. I. If the radius R=l, AB=CA tang , the perpendicular of a right angled triangle is equal tothe base multiplied by the tangent of the angle at the base, theradius being unity. Cor. 2. Since the tangent of an arc is equal to the cotangentof its complement (Art. VI.), the cotangent of B may be sub-stituted in the proportion for tang C, which wilJ giveR : cot B : : CA : AB. THEOREM III. In every rectilineal triangle, the sines of the angles are to eacaoilier as the opposite sides. PLANE TRIGONOMETRY, 239/
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