. Elements of plane and spherical trigonometry . rts are «, next to the adjacent part b, andB next to the adjacent part c. This being understood, Napiers two rulesmay be expressed as follows, carefully observing to use the complementsof the two angles and of the intervening hypotenuse instead of these pojrtsthemselves. I. Rad. X sin. middle part = product of tan. adjacent parts. II. Rad. X sin. middle part =. product of cos. opposite these rules may be comprehended in a single expression, thus rad. sm. mid. =prod. tan. adja. = prod. cos. opp.;and to retain this in the memory we have
. Elements of plane and spherical trigonometry . rts are «, next to the adjacent part b, andB next to the adjacent part c. This being understood, Napiers two rulesmay be expressed as follows, carefully observing to use the complementsof the two angles and of the intervening hypotenuse instead of these pojrtsthemselves. I. Rad. X sin. middle part = product of tan. adjacent parts. II. Rad. X sin. middle part =. product of cos. opposite these rules may be comprehended in a single expression, thus rad. sm. mid. =prod. tan. adja. = prod. cos. opp.;and to retain this in the memory we have only to remember that thevowels in the contractions mid., adja., opp., are the same as those in thecontractions sin., tan., cos., to which they are joined. That these rules comprehend all the equations given above will beseen by taking a, b, c, &c. in succession for the middle part, as in thesubjoined table, keeping in mind the condition just stated, that insteadof A, B. and c, we are to use their * 54 SPHERICAL As in the solution of right-angled triangles two parts are given tofind a third, vre mast ia the application of uNapiers rule choose for themiddle of these three parts that Tvhich causes the other two to becomeeither adjacent parts or opposite parts. EXAMPLES. (53.) 1. In the right-angled triangle ABC are given the two perpen-dicular sides, viz. a = 48=^ 24 16, b = 59° 38 27, to find the hypote-nuse c. Here the hypotenuse being made the middle part the other two will,obviously, be the opposite parts, being separated from the hypotenuseby the intervening angles A, B. Hence by the rulerad X sin. comp. c = cos. a X cos. b;, cos. a cos. b that IS, rad. cos. c — cos. a cos. b .. cos. c = : and as cos. a, rad. cos. b, are both positive, cos. c is positive, and, therefore, c is acute,rad. . 100000000 cos. a 48° 24 16 . 9-8220819 cos. b 59 38 27 . 9-7036515 cos. c 70 23 42 . 9-5257334. 2. In the spherical triangle ABC. right-angled at C, are gi
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