Elements of Plane and Spherical Trigonometry: With Its Applications to the Principles of Navigation and Nautical Astronomy. With the Logarithmic and Trigonometrical Tables |
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Page 18
... suppose the three sides a , b , c , were given , then mul- tiplying the first equation by a , the second by b , and the third by c , we have a2 ab cos . C + ac cos . B b2be cos . A + ba cos . C c2 = ac cos . B + be cos . A ; and ...
... suppose the three sides a , b , c , were given , then mul- tiplying the first equation by a , the second by b , and the third by c , we have a2 ab cos . C + ac cos . B b2be cos . A + ba cos . C c2 = ac cos . B + be cos . A ; and ...
Page 20
... Suppose now that a = 0 , then we have chd.2a = 2 - 2 cos . a . sin . a ' ) 2 . M N • ( 1 ) . As this expression is true for any arc whatever , it is true for the arc a - a ' , so that .... chd.2 ( a -- a ) = 2 — 2 cos . ( a — a ...
... Suppose now that a = 0 , then we have chd.2a = 2 - 2 cos . a . sin . a ' ) 2 . M N • ( 1 ) . As this expression is true for any arc whatever , it is true for the arc a - a ' , so that .... chd.2 ( a -- a ) = 2 — 2 cos . ( a — a ...
Page 40
... Suppose that from the top of a mountain , three miles high , the angle of depression of the remotest visible point of the earth's surface is taken and found to be 2 ° 13 ′ 27 ′′ ; it is required thence to determine the di- ameter of the ...
... Suppose that from the top of a mountain , three miles high , the angle of depression of the remotest visible point of the earth's surface is taken and found to be 2 ° 13 ′ 27 ′′ ; it is required thence to determine the di- ameter of the ...
Page 44
... Suppose the angle of elevation of the top of a steeple to be 40 ° when the observer's eye is level with the bottom , and that from a window 18 feet , directly above the first station , the angle of elevation is found to be 37 ° 30 ...
... Suppose the angle of elevation of the top of a steeple to be 40 ° when the observer's eye is level with the bottom , and that from a window 18 feet , directly above the first station , the angle of elevation is found to be 37 ° 30 ...
Page 51
... Suppose now that A = A , A , & c . then this equation will become ( cos . A + sin . A.√√ = 1 ) " = cos . n A + sin . n A . or , writing the radical with the double sign , ( cos . A ± sin . A. √ = 1 ) " = cos . n A ± sin . nA . √ — 1 ...
... Suppose now that A = A , A , & c . then this equation will become ( cos . A + sin . A.√√ = 1 ) " = cos . n A + sin . n A . or , writing the radical with the double sign , ( cos . A ± sin . A. √ = 1 ) " = cos . n A ± sin . nA . √ — 1 ...
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Common terms and phrases
ABC are given apparent altitude arc BC arith Asin called celestial sphere centre chord circle colatitude comp complement computation correction cosec cosine cotangent course and distance deduced departure determine diff difference of latitude difference of longitude direct course equal equations equinoctial expression find the angle formula given side Greenwich hence horizon hour angle hypotenuse included angle logarithmic measured meridian middle latitude miles Napier's Nautical Almanack negative oblique obtuse opposite angle parallax parallel parallel sailing perpendicular plane sailing plane triangle pole PROBLEM quadrant quantities radius right ascension right-angled triangle rule secant semidiameter ship sin.c sine sine and cosine solution sphere spherical angle spherical triangle spherical trigonometry subtracting supplement tabular line tangent third side three angles three sides triangle ABC trigono trigonometrical lines true altitude values vertical zenith
Popular passages
Page viii - In any plane triangle, the sum of any two sides is to their difference, as the tangent of half the sum of the opposite angles is to the tangent of half their difference.
Page 99 - B . sin c = sin b . sin C cos a = cos b . cos c + sin b . sin c cos b = cos a . cos c + sin a . sin c cos A cos B cos c = cos a . cos b + sin a . sin b . cos C ..2), cotg b . sin c = cos G.
Page 22 - In every plane triangle, the sum of two sides is to their difference as the tangent of half the sum of the angles opposite those sides is to the tangent of half their difference.
Page vi - An Elementary Treatise on Algebra, Theoretical and Practical; with attempts to simplify some of the more difficult parts of the Science, particularly the Demonstration of the Binomial Theorem in its most general form ; the Summation of Infinite Series ; the Solution of Equations of the Higher Order, &c., for the use of Students.
Page 160 - If the zenith distance and declination be of the same name, that is, both north or both south, their sum will be the latitude ; but, if of different names, their difference will be the latitude, of the same name as the greater.
Page 165 - PS' ; the coaltitudes zs, zs', and the hour angle SPS', which measures the interval between the observations ; and the quantity sought is the colatitude ZP. Now, in the triangle PSS , we have given two sides and the included angle to find the third side ss', and one of the remaining angles, say the angle PSS'. In the triangle zss...
Page 129 - To THE TANGENT OF THE COURSE ; So IS THE MERIDIONAL DIFFERENCE OF LATITUDE, To THE DIFFERENCE OF LONGITUDE. By this theorem, the difference of longitude may be calculated, without previously rinding the departure.
Page vi - MICHAEL O'SHANNESSY, AM 1 vol. 8vo. " The volume before us forms the third of an analytical course, which commences with the * Elements of Analytical Geometry.' More elegant t&xtbooks do not exist in the English language, and we trust they will speedily be adopted in our Mathematical Seminaries. The existence of such auxiliaries will, of itself, we hope, prove an inducement to the cultivation of Analytical Science ; for, to the want of such...
Page 69 - The sum of the three sides of a spherical triangle is less than the circumference of a great circle. Let ABC be any spherical triangle ; produce the sides AB, AC, till they meet again in D. The arcs ABD, ACD, will be semicircumferenc.es, since (Prop.
Page 138 - PEP' (Fig. 22,) represent the meridian of the place, Z being the zenith, and HO the horizon ; and let LL' be the apparent path of the sun on the proposed day, cutting the horizon in S. Then the...