*CHAPTER III. CONSTRUCTION OF ANGLES WITH ASSIGNED TRIGONOMETRICAL RATIOS. GENERAL FORMULAE FOR ANGLES WITH GIVEN TRIGONOMETRICAL RATIOS. 22. To construct an angle with a given positive sine or cosine. AB Let AB: BC or be the given sine or cosine, BC BC AB and sinD = sin (90° – A) = cos▲, by Art. (15), Hence, if the sine be given AB and if the cosine be given = BC' A is the required angle. 23. To construct an angle with a given positive tangent or cotangent. (See fig. Art. 22.) AB Let AB: BC or be the given tangent or cotangent. BC Draw BD perpendicular to AB, and = BC, and join AD. 24. To find a general expression for all the angles which Therefore (by Art. 10), if m be zero or a positive or negative integer, the formula 2mπ + a will comprehend all the angles terminated by OP and 2mπ+π— α, or (2m+1) 7 comprehend all the angles terminated by OP'. π- a will Now nπ+(-1)". a, where n is zero or any positive or negative integer, will embrace both the formulae 2mπ + α and a (2m+1) π — a; for if n be an even integer, positive or negative, and = 2m, nπ+(−1)". a becomes 2mπ+α, and if n be an odd integer, positive or negative, and = 2m+1, nπ+(-1)". a becomes (2m+1) π-α. Thus the formula nʊ+(−1)”. a includes all the angles, and no others, in 2m+a and (2m+1)π-a, that is, all the angles which have the same sine as ɑ. The formula n + (-1)". a also includes all the angles which have the same cosecant or coversed sine as a. 25. To find a general expression for all the angles which have a given cosine. Let POM be the least positive angle which has the given cosine, and let a be its circular measure. = Make the P'OM POM, so that the circular measure of P'OM is -α. It is evident from the figure that all the angles terminated by OP and OP' have the given cosine. (See Art. 12.) Therefore (by Art. 10), if n be zero or any positive or negative integer, the formula 2nπ + a comprehends all the angles terminated by OP and 2nπ- a comprehends all the angles terminated by OP'. Thus the formula 2nπ ± a includes all the angles, and no others, which have the same cosine as a. .. cos (2nπ ± a) = cosa. The formula 2nπa also includes all the angles which have the same secant or versed sine as a. 26. To find a general expression for all the angles which have a given tangent. It is evident from the figure that all the angles terminated by OP and OP' have the given tangent. (See Art. 14.) Therefore (by Art. 10), if m be zero or any positive or negative integer, the formula 2m+a comprehends all the angles terminated by OP, and 2mπ++α or (2m+1)+a comprehends all the angles terminated by OP'. The formula në + a evidently comprehends all the angles, and no others, included in these two formulae, that is, all the angles which have the same tangent as a. .. tan (nπ + a) = tan a. The formula n+a also includes all the angles which have the same cotangent as a. |