The elements of plane trigonometry |
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Página 92
John Charles Snowball. From these two cases therefore it appears that the theorem might have been thus enunciated ; If the index be an integer , ( cos + - 1 sin 0 ) " = cos m◊ ± √ − 1 sin me ; If the index be fractional , m ( cose - 1 ...
John Charles Snowball. From these two cases therefore it appears that the theorem might have been thus enunciated ; If the index be an integer , ( cos + - 1 sin 0 ) " = cos m◊ ± √ − 1 sin me ; If the index be fractional , m ( cose - 1 ...
Página 122
... Theorem , Sin 0 and Cose in series ascending by powers of 0 , assuming sin that Ꮎ becomes 1. when I becomes 0 . If sin can be expanded in a series ascending by integral powers of 0 , such a series cannot possibly contain any but odd ...
... Theorem , Sin 0 and Cose in series ascending by powers of 0 , assuming sin that Ꮎ becomes 1. when I becomes 0 . If sin can be expanded in a series ascending by integral powers of 0 , such a series cannot possibly contain any but odd ...
Página 140
... ( 6 + 1 ) = 2 22 42 22 42 62 82 ... π = 2 • 1.3 3.5 5.7 7.9 62 ... π , Wallis's theorem for the determination of in which the successive factors become more and more nearly equal to 1 . APPENDIX I. ON THE LOGARITHMS OF NUMBERS , AND THE 140.
... ( 6 + 1 ) = 2 22 42 22 42 62 82 ... π = 2 • 1.3 3.5 5.7 7.9 62 ... π , Wallis's theorem for the determination of in which the successive factors become more and more nearly equal to 1 . APPENDIX I. ON THE LOGARITHMS OF NUMBERS , AND THE 140.
Página 184
... THEOREM , L sin ( 9 +80 ) = L sin 0 + de L sin 0 . 80+ de L sin 0.- ( 80 ) 2 1.2 + ... 1 cos - Now L sin 0 = 10 + 110 sin 0 ; .. d . L sin 0 : = de L sin 0 ... = - • 10 sin 1 1,10 0 1 1,10 ( cosec ) 2 ; L sin ( 0 + 80 ) − L sin ( 1 ...
... THEOREM , L sin ( 9 +80 ) = L sin 0 + de L sin 0 . 80+ de L sin 0.- ( 80 ) 2 1.2 + ... 1 cos - Now L sin 0 = 10 + 110 sin 0 ; .. d . L sin 0 : = de L sin 0 ... = - • 10 sin 1 1,10 0 1 1,10 ( cosec ) 2 ; L sin ( 0 + 80 ) − L sin ( 1 ...
Página
... THEOREM .. 34. NAPIER'S ANALOGIES , proved from Gauss ' Theorem ....... 23 19 A 35. Cos 4 , sin A A tan " " sin A , in terms of a , b , c ......... 24 a a α " 36. Cos sin tan 2 ' 2 ' 37. NAPIER'S ANALOGIES , independently proved ...
... THEOREM .. 34. NAPIER'S ANALOGIES , proved from Gauss ' Theorem ....... 23 19 A 35. Cos 4 , sin A A tan " " sin A , in terms of a , b , c ......... 24 a a α " 36. Cos sin tan 2 ' 2 ' 37. NAPIER'S ANALOGIES , independently proved ...
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Términos y frases comunes
A₁ AC AC angle containing Asin base C₁ centre circle coefficient cosec cosines decimal determined Diff difference digit equal errors Eucl expression formulæ given goniometric functions greater Hence increment integer last Article less m₁ magnitude mantissa measured Napier's rules nearly negative number of seconds P₁ perpendicular plane triangle planes AOB polar triangle pole polygon positive proved quantity radius Regular Polyhedron right angles right-angled triangle S₁ secants shew shewn Similarly sin A sin sines sines and cosines small angle solid angle sphere spherical excess spherical triangle SPHERICAL TRIGONOMETRY subtended subtraction tabular logarithmic tangents THEOREM triangle ABC triangle whose sides TRIGONOMETRY versin Wherefore π π
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Página 141 - Logarithm of a number to a given base is the index of the power to which the base must be raised to give the number.
Página 10 - Any two sides of a spherical triangle are together greater than the third, and the sum of the three sides is less than the circumference of a great circle.
Página 56 - Mathematics which treats of the solution of plane triangles. In every plane triangle there are six parts : three sides and three angles. When three of these parts are given, one being a side, the remaining parts may be found by computation. The operation of finding the unknown parts is called the solution of the triangle.
Página 57 - ... the three interior angles of a triangle are together equal to two right angles.
Página 12 - The sum of the angles of a spherical triangle is greater than two and less than six right angles ; that is, greater than 180° and less than 540°. (gr). If A'B'C' is the polar triangle of ABC...
Página 143 - The logarithm of a quotient is equal to the logarithm of the dividend minus the logarithm of the divisor.
Página 27 - For example, let it be required to prove the formula sin ( A - B) = sin A. cos В - cos A. sin В from the annexed figure, where ВАC' = A, and C'AD — В ; each angle being greater than a right angle.
Página 5 - ... stretched taut from one point to the other. The string will not lie on the parallel, but will evidently be in a plane which passes through the centre of the sphere. If the two points be on a meridian, the stretched string will lie on the meridian. By angular distance between two points on a sphere is meant the angle subtended at the centre of the sphere by the arc joining the given points. Thus in Fig. 4 the angle NOA is the angular distance of A from N.
Página 6 - If a solid angle be contained by three plane angles, any two of them are together greater than the third. Let the solid angle at A be contained by the three plane angles BAC, CAD, DAB : any two of them shall be together greater than the third.
Página 200 - Three circles whose radii are a, b, c touch each other externally ; prove that the tangents at the points of contact meet in a point whose distance from any one of them is (aba Nj a+b + cj 12.