This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1863 Excerpt: ... a, and all the angles which it includes have the same tangent as a. This formula also determines all the angles which have the same cotangent as a. 69. In Art. 66 we shewed that if a be the least positive angle which has a given sine, the formula nir + (-l)"a includes without excess or defect all the angles which have ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1863 Excerpt: ... a, and all the angles which it includes have the same tangent as a. This formula also determines all the angles which have the same cotangent as a. 69. In Art. 66 we shewed that if a be the least positive angle which has a given sine, the formula nir + (-l)"a includes without excess or defect all the angles which have the same sine as a; it was convenient for distinctness in the demonstration to suppose a the least positive angle which has the given sine. But this restriction can be removed, for we can shew that if ji be any angle, the formula mr + (--will include without excess or defect all the angles which have the same sine as (i. For suppose a to be the least positive angle which has its sine equal to sin/3; then, from what has been proved, we know that /? must be one of the angles included in the formula m7r + (--l)"a where m is zero, or any integer positive or negative. Suppose then /3 = rir +(--l)ra; therefore nir + (-l)"/3 = mr + (-l)"rir + (-iy+ra; and all we have to prove is, that this formula includes without excess or defect all the angles included in the formula mir + (--l)TMa. If n be even the formulae correspond by taking m = n + r; if n be odd, the formulae correspond by taking m = n--r. The formula w + (--l)"/3 will of course also include without excess or defect all the angles which have the same cosecant as (3. 70. Similarly we may shew that if /? be any angle, the angles which have the same cosine or secant or versed sine as /3 will be included without excess or defect in the formula 2wir/3; and that the angles which have the same tangent or cotangent as /3 will be included without excess or defect in the formula nir + j3. 71. Before leaving this part of the subject we will recur to the definitions of the...
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