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. 1851 Excerpt: ...of the octant-lines, AN will be greater than PN, or the cosine will as to magnitude (without regard to sign) be greater than the sine; whereas for any angle, whose bounding line lies between DD ' and either of the octant-lines, PN will be greater than AN, or the sine will be greater than the cosine. Now sin.4 + cos, A ...
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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. 1851 Excerpt: ...of the octant-lines, AN will be greater than PN, or the cosine will as to magnitude (without regard to sign) be greater than the sine; whereas for any angle, whose bounding line lies between DD ' and either of the octant-lines, PN will be greater than AN, or the sine will be greater than the cosine. Now sin.4 + cos, A is positive, (1) when sin is (+) and cos (+), that is, in the whole of the first quadrant; or (2) when sin is (+) and cos--), if sin cos, that is, in the former half of the second quadrant; or (3) when sin is (--) and cos (+), if cos sin, that is, in the latter half of the fourth quadrant: collecting which results, we may say that sin A + cos.4 = + V(l + sin 2.4), when A is an angle in the first quadrant, or in either of the two adjacent half-quadrants. Similarly, sin.4-f-cos A =---J(1 + sm2A)y when A is an angle in the third quadrant, or in either of the two adjacent half-quadrants. Again, smA--cos.4 is positive, (1) when sin is (+) and cos (--), that is, in the whole of the second quadrant; or (2) when sin is (+) and cos (+), if sin cos, that is, in the latter half of the first quadrant; or (3) when sin is (--) and cos (--), if cos sin, that is, in the former half of the third quadrant: collecting which results, we may say that ein A--cos A = + V(l--sin21), when A is an angle in the second quadrant, or in either of the two adjacent half-quadrants. Similarly, sinA--cos.4 =--V(l--sin2.4), when A is an angle of the fourth quadrant, or in either of the two adjacent half-quadrants. It will be seen that each of these formulae holds good for 180 together, between alternate octant-lines, and that any pair of them, (namely, one for sinL + cos.4, and one for sin.4--cos.4, ) holds good for 90 together, between successive lines of octants: that is, ...
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