. A treatise on plane and spherical trigonometry, and its applications to astronomy and geodesy, with numerous examples . 40. An Expression for All Angles with a Given Tangent a. — Let 0 be the centre of the unitcircle. Draw AT, touching the circleat A, and take AT so that its measureis a. Join OT, cutting the circle at ~PfiC-and P\ Then it is clear from the figure thatthe only angles which have the tan-gent equal to a are a and it + a, and the angles which differ. 42 PLANE TRIGONOMETRY. from either by a multiple of four right angles. Hence if 6be the general value of the required angle, we ha
. A treatise on plane and spherical trigonometry, and its applications to astronomy and geodesy, with numerous examples . 40. An Expression for All Angles with a Given Tangent a. — Let 0 be the centre of the unitcircle. Draw AT, touching the circleat A, and take AT so that its measureis a. Join OT, cutting the circle at ~PfiC-and P\ Then it is clear from the figure thatthe only angles which have the tan-gent equal to a are a and it + a, and the angles which differ. 42 PLANE TRIGONOMETRY. from either by a multiple of four right angles. Hence if 6be the general value of the required angle, we have 6 — 2mr-{-a, and 2ft7r + 7r+a (1) Also, the only negative angles which have the tangentequal to a are — (?r — a), and — (2?r — a), and the angleswhich differ from either by a multiple of four right anglestaken negatively ; that is, we have 0 = 27nr-(7r-a), and 2nir - (2tt - a), . (2) where n is zero or any negative integer. Xow the angles in (1) and (2) may be arranged thus: 27iir + a, (2n + l)7r + a, (2n — l)ir + a, (2n — 2) ?r + a, all of which, and no others, are included in the formula 0 = 7iw + a, (3) where n is zero, or any positive or negative integer. There-fore (3) is the general expression for all angles which havea given tangent. Note. — The same formula determines all the angles which have the same cotan-gent as a. EXAMPLES. 1. Find six angles between — 4 right angles and + 8right angles which satisfy the equation sinA = s
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