The elements of plane trigonometry |
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Página 3
... Similarly , the ordinates of P and P1 measured in the direction of XAX ' are negative and positive respectively ; and their ordinates measured in the direction of YAY ' are negative in both cases . D F C E ' 4. DEF . If a straight line ...
... Similarly , the ordinates of P and P1 measured in the direction of XAX ' are negative and positive respectively ; and their ordinates measured in the direction of YAY ' are negative in both cases . D F C E ' 4. DEF . If a straight line ...
Página 10
... similarly it may be shewn that cos 4 , tan A , sec A ... are invariable quantities so long as the magnitude of A remains unaltered . Hence , if any of the quantities sin A , cos A , tan A , sec A ... be given , the angle A may be ...
... similarly it may be shewn that cos 4 , tan A , sec A ... are invariable quantities so long as the magnitude of A remains unaltered . Hence , if any of the quantities sin A , cos A , tan A , sec A ... be given , the angle A may be ...
Página 15
... Similarly , sin A = sin ( 180 ° - A ) Art . 22. ( 1 ) . = sin { 4n.90o + ( 180 ° – A ) } - = sin { ( 4n + 2 ) 90 ° – 4 } .... - In like manner we get from ( 2 ) and ( 4 ) of Art . 22 , sin A = - sin { ( 4n + 2 ) 90o + A } . sin A ...
... Similarly , sin A = sin ( 180 ° - A ) Art . 22. ( 1 ) . = sin { 4n.90o + ( 180 ° – A ) } - = sin { ( 4n + 2 ) 90 ° – 4 } .... - In like manner we get from ( 2 ) and ( 4 ) of Art . 22 , sin A = - sin { ( 4n + 2 ) 90o + A } . sin A ...
Página 16
... Similarly it might be proved that , sec A = sec ( 4n . 90 ° + A ) , or = sec { ( 4n + 2 ) . 90o - - A } , or = -sec { ( 4n + 2 ) . 90 ° + A } , or = sec ( 4n . 90 ° - A ) . 26. From Arts . 16 , 22 , 23 , we collect that sin A = cos ( 90 ...
... Similarly it might be proved that , sec A = sec ( 4n . 90 ° + A ) , or = sec { ( 4n + 2 ) . 90o - - A } , or = -sec { ( 4n + 2 ) . 90 ° + A } , or = sec ( 4n . 90 ° - A ) . 26. From Arts . 16 , 22 , 23 , we collect that sin A = cos ( 90 ...
Página 21
... Similarly , for angles between 45 ° and 90o , it may be shewn that the cosine is less than the sine . 32. To find the sines , cosines , and tangents , of 45o , 30 ° , and 60o . ( 1 ) ( Fig . Art . 27. ) Let NAC = 45 ° ; .. LNCA = 90 ...
... Similarly , for angles between 45 ° and 90o , it may be shewn that the cosine is less than the sine . 32. To find the sines , cosines , and tangents , of 45o , 30 ° , and 60o . ( 1 ) ( Fig . Art . 27. ) Let NAC = 45 ° ; .. LNCA = 90 ...
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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.