The elements of plane trigonometry |
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Página v
... x + 2 cos 0 , then a 2 V - 1 . sin 0 89 x x 106. DEMOIVRE'S THEOREM . 89 1 107. If + = 2 cos 0 , then a2 + X xn 2 cos n0 , and " -- Xon 1 2 √1 . sin no .... 92 ART . 108. To express any positive integral power of INDEX . V.
... x + 2 cos 0 , then a 2 V - 1 . sin 0 89 x x 106. DEMOIVRE'S THEOREM . 89 1 107. If + = 2 cos 0 , then a2 + X xn 2 cos n0 , and " -- Xon 1 2 √1 . sin no .... 92 ART . 108. To express any positive integral power of INDEX . V.
Página vii
... THEOREM ......... 121 APPENDIX I. ON THE LOGARITHMS OF NUMBERS , AND THE CONSTRUCTION AND USE OF LOGARITHMIC TABLES OF NUMBERS . 1. Definitions ........ 141 2. Ln = a " , La = 1 , 1,1-0 , m1g " = ŋlog m 142 3. Having given tables of ...
... THEOREM ......... 121 APPENDIX I. ON THE LOGARITHMS OF NUMBERS , AND THE CONSTRUCTION AND USE OF LOGARITHMIC TABLES OF NUMBERS . 1. Definitions ........ 141 2. Ln = a " , La = 1 , 1,1-0 , m1g " = ŋlog m 142 3. Having given tables of ...
Página 89
... THEOREM . To shew that for any value of m , ( cos +/- 1 sin 0 ) " = cos m 0 ± √√√ - 1 sin m0 . ( Cos 0-1 sin 0 ) . ( cos +/- 1 sin 0 ) = 0 ± ( cos 0 ) — ( sin ( ) 2 ± √ − 1 . 2 sin 0 . cos 0 . Or , ( cos + v 1 sin 0 ) = cos 20 ...
... THEOREM . To shew that for any value of m , ( cos +/- 1 sin 0 ) " = cos m 0 ± √√√ - 1 sin m0 . ( Cos 0-1 sin 0 ) . ( cos +/- 1 sin 0 ) = 0 ± ( cos 0 ) — ( sin ( ) 2 ± √ − 1 . 2 sin 0 . cos 0 . Or , ( cos + v 1 sin 0 ) = cos 20 ...
Página 90
... ) 2 + ( sin 0 ) 2 cos 0-1 sin ✪ ( cos - 1 sin 0 ) " , by actual division ; m m = cos me = √ - 1 sin m0 ,干 9 = cos ( -m0 ) ± √ - 1 sin ( - m 0 ) ; which proves the theorem for negative indices . Again , ( cos & ± √1 sin ) " 90.
... ) 2 + ( sin 0 ) 2 cos 0-1 sin ✪ ( cos - 1 sin 0 ) " , by actual division ; m m = cos me = √ - 1 sin m0 ,干 9 = cos ( -m0 ) ± √ - 1 sin ( - m 0 ) ; which proves the theorem for negative indices . Again , ( cos & ± √1 sin ) " 90.
Página 91
... theorem for fractional indices . COR . By the theorem just proved , - I n m - n -0 ; ( cos + √ sin p ) " = cos m ± √ - 1 sin mp , m being positive or negative , whole or fractional . Let 2p + 0 , where p is any integer ; ф = 2рп 1 ...
... theorem for fractional indices . COR . By the theorem just proved , - I n m - n -0 ; ( cos + √ sin p ) " = cos m ± √ - 1 sin mp , m being positive or negative , whole or fractional . Let 2p + 0 , where p is any integer ; ф = 2рп 1 ...
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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
Pasajes populares
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.