A Treatise of Algebra Wherein the Principles Are Demonstrated to Which is a
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And that the rectangle of the tangent and co-tangent is equal to the square of the radius. Of Plane Trigonometry* 245 PROPOSITION III. The co-sine CF of an arch AE being given ; to find the sine and cosine of half that arch* From the two extremities of the diameter AB draw the subtenses AE and BE ; and let CQ bisect the arch AE in Q and its chord (perpendicularly) in D ; then, since the angle BEA is a right one- (by Euc. 31, 3. ), the triangles ABE and ^^^^ ^ ADC are similar ; and, therefore, A...C being = I AB, AD must be = I AE, and CD = \ BE : b ut AE is = VA B X AF, an d BE = VAB X BF ; therefore AD = |VAB_ K¥ = \ /|AC x A F = the sine ? of ^ CD = IVAB X iiF = VIAC X BF = the co-sine 5 ^E. Hence it is evident, that the sine of the half of any- arch is a mean proportional between the half radius and the versed-sine of the whole arch ; and its co-sine a mean proportional between half the radius and the versed-sine of the supplement of the same arch. PROPOSITION IV. The sine AD, and co-sine CD, of an arch AQ being given ; to find EF, the sine of the double of that arch.
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