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Cor. 1. Hence, likewise, the rectangle under the radius and the sine of the difference of two arcs, is equal to the difference of the rectangles under their alternate sines and cosines; or R sinAC'-sin AB cosBC—cos AB sinBC.

Cor. 2. If the two arcs A and B be equal, it is obvious that R sin2A=sinA2cos A.

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Cor. 4. Let 2A=C, and, by the second corollary,
It sinC-sin;C 2cos; C.

PROP. II. THEOR.

The rectangle under the radius and the cosine of the sum of two arcs, is equal to the difference of the rectangles under their respective cosines and sines.

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Cor. 1. Hence, likewise, the rectangle under the radius and the cosine of the difference of two arcs is equal to the sum of the rectangles under their respective cosines and sines; or Ricos AC/=cos AB cosBC+sinAB sin BC.

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Of the equidifferent arcs, the rectangle under the radius and the sum of the sines of the extremes, is equal to twice the rectangle under the cosine of the common difference and the sine of the mean arc.

Let A–B, A, and A+B represent three arcs increasing by the difference B; then

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The property is easily deduced by combining the preceding theorems; but it s will be more easily B p perceived, by refer- H jo N o ring immediately to § C the original figure.

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Cor. 1. Hence, likewise, of three equidifferent arcs, the rectangle under the radius and the difference of the sines of the extremes, is equal to twice the rectangle under the

sine of the common difference and the cosine of the mean

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Cor. 6. Produce CE to the circumference, join C'I meeting the production of FG in K, and join OK. Since FK is parallel to CI and bisects CC, it likewise bisects IC'; and hence OK is perpendicular to KC, which is, therefore, the sine of half the arc IAC", or of half the sum of the arcs AC and AC", as CF is the sine of half their difference. But (II.21.E.)IC*-CC”=IC.2C'E', or C'K*–CF=CE.C'E'; consequently sin”AB—sin”BC=sinAC sinAC, or, employing the general notation,

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Scholium. By help of this proposition, the sines and cosines of multiple arcs are easily determined; but the expressions for them will become simpler, if, as in cor. 2. the radius be supposed equal to unit. For A, 2A and 3A being three equidifferent arcs, sinA+sin3A=2CosA sin2A=2cos/A 2CosAsin A, or sin3A-4cosA*.sin A–sin A; and cosA+cos3A=2cosA.coszA-2cosA(2cos/A*–1)= 4cos A*–2cos A, or cos3A=4cos A*–3cos A.

Again, since 2A, 3A, and 4A are equidifferent arcs, sin2A+sin&A=2cosA sin3A=8cosA' sin A–2cos A sin A, or sin&A=8cosA” siná–4cosA sinA; cos2A+cos4A=2cosA.cos3A=2cosA(4cos A*–3cosA), or cos4A = 8CosA* 8cos A*-H 1. In like manner, assuming the equidifferent arcs 3A, 4A, 5A, the sine and cosine of 5A are found ; and this mode of procedure may be continually repeated. To abridge the notation, however, it will be proper to express the sine and the cosine of the arc a, by s and c. The results are thus expressed in a tabular form: .

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