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. 1883 Excerpt: ...us with the sines and cosines of all angles at intervals of 1' from 45 to 90. METHODS OF VERIFICATION. 124. In calculating the s-iue and cosine of 1' in Art. 120, we obtained the results correct to ten decimal places; and, in deriving from them the sines and cosines of the remaining angles up to 30, we performed 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. 1883 Excerpt: ...us with the sines and cosines of all angles at intervals of 1' from 45 to 90. METHODS OF VERIFICATION. 124. In calculating the s-iue and cosine of 1' in Art. 120, we obtained the results correct to ten decimal places; and, in deriving from them the sines and cosines of the remaining angles up to 30, we performed a great number of operations with these originiil approximate numbers. It is not likely, then, that all our sines and cosines up to 30 will be obtained correct to ten decimal places, and it is important to know to how many places they may be depended upon as accurate. It is also important to be able to verify the work from time to time to ascertain if any error has been made in the calculations, since any mistake made in obtaining the value of one of the functions would be repeated to the end of the work. For this purpose we may compare our results for the angles 15, 18, 22 30', 30, 36, and 45, calculated in the manner explained, with the sines and cosines of the same angles as derived in Arts. 17, 18, 74, 75, 76, 77. It is also easy, by aid of the formulae of Art. 69, to derive from the functions of 15 the sines and cosines of the angles 7 30', 3 45'; from the functions of 18, the sines and cosines of 9, 4 30', .; from the functions of 2230', the sines and cosines of 11 15', 537'30,.". In this way other portions of the calculated table may be verified. 125. Another method of verification, known as Euler's, is proved as follows: By Art. 72, sin (36+A)-sin (36-l) = 2cos36smt Also, sin (72+l)-sin (72-l) = 2cos72sin = V/51sinj (Art. 76) Subtracting these results, we have sin (36 +.4)-sin (36-l)-sin (72+l) + sin ( 72--A) = sin A (63) which is Euler's Formula. If to A in this formula we give any value, say 10, we have sin46-sm26o-sin82o+sin62=sinl0 ...
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