Higher Geometry and Trigonometry: Being the Third Part of a Series on Elementary and Higher Geometry, Trigonomentary and Mensuration : Containing Many Valuable Discoveries and Improvements in Mathematical Science, Especially in Relation to the Quadrature of the Circle, and Some Other Curves, as Well as the Cubature of Certain Curvilinear Solids : Designed as a Text-book for Collegiate and Academic Instruction, and as a Practical Compendium of Mensuration |
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Page 35
... half the arc . Take any arc AQ , subtending at the centre of the circle the angle ACQ = 0 . Draw the straight line CP bisecting the angle ACQ . Join A , Q ; from P let fall PS perpen- dicular on CA. Since CP bisects ACQ , the vertical ...
... half the arc . Take any arc AQ , subtending at the centre of the circle the angle ACQ = 0 . Draw the straight line CP bisecting the angle ACQ . Join A , Q ; from P let fall PS perpen- dicular on CA. Since CP bisects ACQ , the vertical ...
Page 49
... half a given angle . By formula ( 13 ) : cos . 20 = 1 2 sin . θ - For substitute the above becomes , 2 8 cos . 2 --- - 2 sin.1 2 2 Or , cos . 8 = 1-2 sin . * .. 2 sin . ' 8 - cos . A 2 8 sin . ༡ == COS . 2 ( j ) That is , the sine of ...
... half a given angle . By formula ( 13 ) : cos . 20 = 1 2 sin . θ - For substitute the above becomes , 2 8 cos . 2 --- - 2 sin.1 2 2 Or , cos . 8 = 1-2 sin . * .. 2 sin . ' 8 - cos . A 2 8 sin . ༡ == COS . 2 ( j ) That is , the sine of ...
Page 50
... half a given angle or arc is equal to the square root of 1 minus the cosine of the arc divided by two . To determine the cosine of half a given angle . By formal ( h2 ) : cos . 28 = 2 cos.26-1 ; the above becomes , 8 For substitute 2 8 ...
... half a given angle or arc is equal to the square root of 1 minus the cosine of the arc divided by two . To determine the cosine of half a given angle . By formal ( h2 ) : cos . 28 = 2 cos.26-1 ; the above becomes , 8 For substitute 2 8 ...
Page 57
... half the arc ; therefore , 2 sin . 45 ° chord 90 ° 4 sin . ' 45 ° -chord " 90 ° = 2 , by Equation ( 1 ) ; 1 sin . 45 ° = √2 Again , by table I. : sin . ' + cos . = 1 cos . 45 ° -1 - sin . ' 45 ° 1 = 1 2 1 = 2 1 cos . 45 ° = = sin . 45 ...
... half the arc ; therefore , 2 sin . 45 ° chord 90 ° 4 sin . ' 45 ° -chord " 90 ° = 2 , by Equation ( 1 ) ; 1 sin . 45 ° = √2 Again , by table I. : sin . ' + cos . = 1 cos . 45 ° -1 - sin . ' 45 ° 1 = 1 2 1 = 2 1 cos . 45 ° = = sin . 45 ...
Page 68
... half the sum of the angles opposite to them , is to the tangent of half their difference . Let ABC be any plane triangle , then , by Proposition II b a + b - sin . A sin . B sin . A + sin . B = A B a sin . A - sin . B But , by ...
... half the sum of the angles opposite to them , is to the tangent of half their difference . Let ABC be any plane triangle , then , by Proposition II b a + b - sin . A sin . B sin . A + sin . B = A B a sin . A - sin . B But , by ...
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abscissa altitude arithmetical progression axes base bisected chord circle circular circular segment circumference cone conjugate construction convex surface corresponding cosec cosine cylinder described diameter distance divided draw drawn ellipse equal to half equation expression feet formed formula frustum Geom geometrical given height hence hyperbola inches infinite series latus rectum length logarithm major axis multiplied opposite ordinates parabola parallel parallelogram perpendicular plane portion prism Prop PROPOSITION pyramid quadrant quadrature quantity radii radius ratio rectangle represent revoloidal surface right angles Scholium sector segment sides similar similar triangles sine solidity specific gravity sphere spherical triangle spheroid spindle square straight line tangent THEOREM tion trian triangle ABC trigonometrical ungula versed sine vertex vertical virtual centre whence
Popular passages
Page 81 - The logarithm of any power of a number is equal to the logarithm of the number multiplied by the exponent of the power.
Page 81 - N .-. by definition, x — x" is the logarithm of ^ ; that is to say, The logarithm of a fraction, or of the quotient of two numbers, is equal to the logarithm of the numerator minus the logarithm of the denominator. III. Raise both members of equation (1) to the nth power. N"=a".
Page 68 - In the same way it may be proved that a : b : : sin. A : sin. B, and these two proportions may be written a : 6 : c : : sin. A : sin. B : sin. C. THEOREM III. t8. 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. By Theorem II. we have a : b : : sin. A : sin. B.
Page 7 - The radius of a sphere is a straight line, drawn from the centre to any point of the surface ; the diameter, or axis, is a line passing through this centre, and terminated on both sides by the surface.
Page 138 - 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 8 - A spherical triangle is a portion of the surface of a sphere, bounded by three arcs of great circles.
Page 27 - The circumference of every circle is supposed to be divided into 360 equal parts, called degrees...
Page 78 - In a system of logarithms all numbers are considered as the powers of some one number, arbitrarily chosen, which is called the base of the system, and the exponent of that power of the base which is equal to any given number, is called the logarithm of that number. Thus, if a be the base of a system of logarithms, N any number, and x such that N = a* then x is called the logarithm of N in the system whose base is a.