Elements of Plane and Spherical Trigonometry: With Its Applications to the Principles of Navigation and Nautical Astronomy. With the Logarithmic and Trigonometrical Tables |
From inside the book
Page viii
Proof that the angles A , B , C , of a plane triangle have this remarkable property ,
viz . tan . A + tan . B + tan . C = tan . A tan . ... General expressio for sin . n A and
cos . n A deduced from De Moivre's formula ib . Various formulas for double arcs
...
Proof that the angles A , B , C , of a plane triangle have this remarkable property ,
viz . tan . A + tan . B + tan . C = tan . A tan . ... General expressio for sin . n A and
cos . n A deduced from De Moivre's formula ib . Various formulas for double arcs
...
Page 16
10 : AC 288 2.4593925 :: cos . ... A = 54 ° 36 ' 14 " , C = 35 ° 23 ' 46 " , AC = 92 •
23 . ... Oblique - Angled Triangles . с Let ABC be any plane triangle , and let us
denote the angles by the capital letters , A , B , C , at their vertices , and the sides
opposite to them by the small letters a , b , c . ... 5 , ) CD b sin . A. In like manner
the sine of B to the radius a , is the same line CD , whose value , therefore , in
term of ...
10 : AC 288 2.4593925 :: cos . ... A = 54 ° 36 ' 14 " , C = 35 ° 23 ' 46 " , AC = 92 •
23 . ... Oblique - Angled Triangles . с Let ABC be any plane triangle , and let us
denote the angles by the capital letters , A , B , C , at their vertices , and the sides
opposite to them by the small letters a , b , c . ... 5 , ) CD b sin . A. In like manner
the sine of B to the radius a , is the same line CD , whose value , therefore , in
term of ...
Page 17
B = bsin . A .. sin . A sin . B 6 This equation immediately furnishes us with an
important rule , which may be expressed as follows . ... The cosine of A , to the
radius b , is the line AD ; and , therefore , AD , in terms of the trigonometrical
cosine of A , is AD = b cos . ... for the side AB the expression c = a cos . B + bcos .
A ; in which the proper signs of the cosines are supposed to be involved in their
expressions .
B = bsin . A .. sin . A sin . B 6 This equation immediately furnishes us with an
important rule , which may be expressed as follows . ... The cosine of A , to the
radius b , is the line AD ; and , therefore , AD , in terms of the trigonometrical
cosine of A , is AD = b cos . ... for the side AB the expression c = a cos . B + bcos .
A ; in which the proper signs of the cosines are supposed to be involved in their
expressions .
Page 21
-sin.acos . a .... ( 11 ) . Lastly , putting ( a ta ' ) for a , in the equations ( I ) and ( II ) ,
we have , cos . a = cos . ( a ta ' ) cos . ' + sin . ... ( ata ) and cos . ( a + a ) , multiply
the first by sin . c ' , the second by cos . ' , and add , and we thus get sin . ( a ta ' ) =
sin . a cos . a ' + sin . a'cos . a . ... ( a - « ' ) sin . ( a ta ' ) — sin . ( a - ) sin . a cos . a '
cos , a sin . a ( art . 8 ) tan , a If , therefore , we put ata = A , a - a = B .. a = i ( A ...
-sin.acos . a .... ( 11 ) . Lastly , putting ( a ta ' ) for a , in the equations ( I ) and ( II ) ,
we have , cos . a = cos . ( a ta ' ) cos . ' + sin . ... ( ata ) and cos . ( a + a ) , multiply
the first by sin . c ' , the second by cos . ' , and add , and we thus get sin . ( a ta ' ) =
sin . a cos . a ' + sin . a'cos . a . ... ( a - « ' ) sin . ( a ta ' ) — sin . ( a - ) sin . a cos . a '
cos , a sin . a ( art . 8 ) tan , a If , therefore , we put ata = A , a - a = B .. a = i ( A ...
Page 22
A + sin . B : sin . A -sin . B :: a + b : a - b ; consequently , from the equation above ,
a a + b tan . & ( A + B ) - b tan . ... and , therefore , 62 + -a ' = ( b + c + a ) ( b + ( - a )
+ 2 bc ; that expression may be put under the form ( b + c + a ) ( b + c - a ) cos .
A + sin . B : sin . A -sin . B :: a + b : a - b ; consequently , from the equation above ,
a a + b tan . & ( A + B ) - b tan . ... and , therefore , 62 + -a ' = ( b + c + a ) ( b + ( - a )
+ 2 bc ; that expression may be put under the form ( b + c + a ) ( b + c - a ) cos .
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Elements of Plane and Spherical Trigonometry: With Its Applications to the ... John Radford Young No preview available - 2015 |
Common terms and phrases
added altitude angle apparent applied arith becomes called celestial centre circle comp computation consequently considered correction corresponding cosine course declination deduced departure determine diff difference difference of latitude direction distance dividing drawn east equal equations EXAMPLES expression find the angle follows formula given greater hence horizon hour known latitude less logarithmic longitude means measured meridian method middle miles multiplying Nautical negative object observed obtained opposite parallel passing perpendicular plane plane triangle pole positive PROBLEM proportion quadrant quantities radius remaining remarked represent respectively right-angled triangle rule sailing ship sides signs sine solution sphere spherical triangle substituting subtracting supplement Suppose surface taken tangent third three sides triangle ABC trigonometrical true values
Popular passages
Page vi - 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 97 - 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 20 - 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 iv - 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 158 - 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 163 - 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 127 - 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 iv - 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 67 - 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 136 - 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...