The Elements of Trigonometry

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. 1904 Excerpt: ... this ground measure the angle of elevation a of the top of the tower. Measure off a distance AB (= a) from A directly toward the foot of A the tower, and at B measure the angle of elevation ft. To find the unknown height x of the tower, we have to connect it with the measured length a. This is best done as follows: (2), and, from the triangle PAB, we have PB sin PAB sin a a sin BPA sin (B-a)' since Lrpa = Lqrp-Lqap = B-a. From (1) and (2), by multiplication, we have x sin a sin B a sin (B--a)' sin a sin B i.e. x = a--.--TTj;. sin (B--a) The height x is therefore given in a form suitable for logarithmic calculation. Numerical Example. If a = 100 feet, a = 30, and /3 = 60, then lOOlOOxf = 86-6 feet, sin oU & 128. It is often not convenient to measure AB directly towards Q. Measure therefore AB in any other suitable direction on the horizontal ground, and at A measure the angle of elevation a of P, and also the angle PAR (= /3). At B measure the angle PBA A (=y)-a In the triangle PAB, we have then' B L APR = 180-L PAB-Lpba = 180-(B + y). AP va.PBA _ sin y Hence = sm i?P = sin (/3 + y)' From the triangle PAQ, we have. sin a sin y x = AP sin a = a---77;. sin (p4y) Hence x is found by an expression suitable for logarithmic calculation. 129. Instead of the measurements in the foregoing article the elevations, PAQ and PBQ, at the points A and B might be taken and the distance AB and the angle QAB measured. Unless, however, the angle QAB. is a right angle the solution is not in general easy or suitable for logarithmic calculation. When QAB is a right angle, the method of procedure will be easily seen from the following example. A man observes that at a point due south of a certain tower its angle of elevation is 60; he. then walks 300 feet due west on a horizon...