The elements of plane trigonometry |
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Página 40
... give us , COS 2 A - = 1 2 sin A. sin A = 1 - 2 ( sin A ) 2 . cos 3 A = cos A - 2 sin A. sin 2 A = cos A - 2 sin A. 2 sin A. cos A = cos 4-4 cos 4. { 1- ( cos A ) } = 4 ( cos A ) 3 - 3 cos A. 58. To find the sines and cosines of 18 ° 40.
... give us , COS 2 A - = 1 2 sin A. sin A = 1 - 2 ( sin A ) 2 . cos 3 A = cos A - 2 sin A. sin 2 A = cos A - 2 sin A. 2 sin A. cos A = cos 4-4 cos 4. { 1- ( cos A ) } = 4 ( cos A ) 3 - 3 cos A. 58. To find the sines and cosines of 18 ° 40.
Página 62
... , 2 , 3 , ... successively , gives the series of angles + 4 , A A A A 360 ° ± 2 × 360 ° ± 3 × 360 ° ± 2 2 whose cosines are all of the same magnitude as the cosine of A A £ or of + • 2 Second . Let n be odd , and = 2m 62.
... , 2 , 3 , ... successively , gives the series of angles + 4 , A A A A 360 ° ± 2 × 360 ° ± 3 × 360 ° ± 2 2 whose cosines are all of the same magnitude as the cosine of A A £ or of + • 2 Second . Let n be odd , and = 2m 62.
Página 63
... gives the series of angles 2 180 ° ± 360 ° + ( 1804 ) , 2.360 ° + ( : 180 ° ± 4 ) > 2 ..... whose cosines are all of the same magnitude as the cosine of A A 180 ° ± 2 which = cos ( 180 ° -4 ) = cos 4 - . 2 2 Wherefore the first member ...
... gives the series of angles 2 180 ° ± 360 ° + ( 1804 ) , 2.360 ° + ( : 180 ° ± 4 ) > 2 ..... whose cosines are all of the same magnitude as the cosine of A A 180 ° ± 2 which = cos ( 180 ° -4 ) = cos 4 - . 2 2 Wherefore the first member ...
Página 66
... give the result with the greatest practical degree of accuracy . Thus in the last case , if b be very small , the angle A is nearly a right angle , and the increment of sin A correspond- ing to a small given increment d △ of the angle ...
... give the result with the greatest practical degree of accuracy . Thus in the last case , if b be very small , the angle A is nearly a right angle , and the increment of sin A correspond- ing to a small given increment d △ of the angle ...
Página 70
... give a with more exactness than the last Article does . a2 = b2 + c2 −2bc cos A - = 2 b2 + c2 - 2bc . 2 { 2 ( cos - 1 ) -1 } COS ( b2 + c2 + 2bc ) − 4bc . = ( b + c ) 2 . { 1 - ( ( 2 be √bc b + c COS COS 2 Art . 39. ( 2 ) . Now ( √b ...
... give a with more exactness than the last Article does . a2 = b2 + c2 −2bc cos A - = 2 b2 + c2 - 2bc . 2 { 2 ( cos - 1 ) -1 } COS ( b2 + c2 + 2bc ) − 4bc . = ( b + c ) 2 . { 1 - ( ( 2 be √bc b + c COS COS 2 Art . 39. ( 2 ) . Now ( √b ...
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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.