Trigonometry, Plane and Spherical: With the Construction and Application of LogarithmsKimber and Conrad, 1810 - 125 |
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Strona 15
... co - tangent is known . 4. EF : EC ( CD ) :: CD : CH ; whence the co - secant is known . Hence it appears , 1. That the tangent is a fourth proportional to the co - sine , the sine , and ... co - tang . OF SINES , TANGENTS , AND SECANTS . 15.
... co - tangent is known . 4. EF : EC ( CD ) :: CD : CH ; whence the co - secant is known . Hence it appears , 1. That the tangent is a fourth proportional to the co - sine , the sine , and ... co - tang . OF SINES , TANGENTS , AND SECANTS . 15.
Strona 16
... co - tang . P : co - tang . Q : : tang . Q : tang . P ; or as co - tang . P : tang . Q : co - tang . Q : tang . P ( by 16. 6. ) . PROP . II . If there be three equidifferent arches AB , AC , AD ( fig . 10. ) , it will be , as radius is ...
... co - tang . P : co - tang . Q : : tang . Q : tang . P ; or as co - tang . P : tang . Q : co - tang . Q : tang . P ( by 16. 6. ) . PROP . II . If there be three equidifferent arches AB , AC , AD ( fig . 10. ) , it will be , as radius is ...
Strona 29
... co - sine of EOF ( or BAC ) :: tang . AC : tang . AB . 2. E. D. COROLLARY 1 . Hence it follows , that the sines of the angles of any oblique spherical triangles ADC are to one another , directly , as the sines of the opposite sides ...
... co - sine of EOF ( or BAC ) :: tang . AC : tang . AB . 2. E. D. COROLLARY 1 . Hence it follows , that the sines of the angles of any oblique spherical triangles ADC are to one another , directly , as the sines of the opposite sides ...
Strona 30
... co - sine ACB : co - sine DCB :: tang . DC : tang . AC . THEOREM II . any right - angled spherical triangle ( ABC ) ( fig . 11. ) it will be , as radius is to the co - sine of one leg , so is the co- sine of the other leg to the co ...
... co - sine ACB : co - sine DCB :: tang . DC : tang . AC . THEOREM II . any right - angled spherical triangle ( ABC ) ( fig . 11. ) it will be , as radius is to the co - sine of one leg , so is the co- sine of the other leg to the co ...
Strona 31
... co- sine of the adjacent leg to the co - sine of the opposite angle . DEMONSTRATION . Let CEF be as in the preceding proposition ; then , by Theor . 1. Case 1. it will be , radius ... co - tang . BC : co - tang . SPHERICAL TRIGONOMETRY . 31.
... co- sine of the adjacent leg to the co - sine of the opposite angle . DEMONSTRATION . Let CEF be as in the preceding proposition ; then , by Theor . 1. Case 1. it will be , radius ... co - tang . BC : co - tang . SPHERICAL TRIGONOMETRY . 31.
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ABDP AC by Theor adjacent angle arch bisecting chord circle passing co-sine AC co-tangent of half common logarithm common section Comp describe the circle E. D. COROLLARY E. D. PROP equal to half extremes gent given angle given circle given point half the difference half the sum half the vertical Hence hyperbolic logarithm hypothenuse inclination intersect leg BC line of measures original circle parallel perpendicular plane of projection plane triangle ABC primitive PROB produced projected circle projected pole projecting point radius rectangle right line right-angled spherical triangle SCHOLIUM secant semi-tangents sides similar triangles sine 59 sine AC sine of half sphere spherical angle SPHERICAL PROJECTIONS spherical triangle ABC sum or difference tangent of half THEOREM THOMAS SIMPSON triangle ABC fig versed sine vertical angle whence
Popularne fragmenty
Strona 69 - TO THEIR DIFFERENCE ; So IS THE TANGENT OF HALF THE SUM OF THE OPPOSITE ANGLES', To THE TANGENT OF HALF THEIR DIFFERENCE.
Strona 79 - ... projection is that of a meridian, or one parallel thereto, and the point of sight is assumed at an infinite distance on a line normal to the plane of projection and passing through the center of the sphere. A circle which is parallel to the plane of projection is projected into an equal circle, a circle perpendicular to the plane of projection is projected into a right line equal in length to the diameter of the projected circle; a circle in any other position is projected into an ellipse, whose...
Strona 25 - The cotangent of half the sum of the angles at the base, Is to the tangent of half their difference...
Strona 28 - The rectangle of the radius, and sine of the middle part, is equal to the rectangle of the tangents of the two EXTREMES CONJUNCT, and to that of the cosines of the two EXTREMES DISJUNCT.
Strona 7 - If the sine of the mean of three equidifferent arcs' dius being unity) be multiplied into twice the cosine of the common difference, and the sine of either extreme be deducted from the product, the remainder will be the sine of the other extreme. (B.) The sine of any arc above 60°, is equal to the sine of another arc as much below 60°, together with the sine of its excess above 60°. Remark. From this latter proposition, the sines below 60° being known, those...
Strona 28 - In a right angled spherical triangle, the rectangle under the radius and the sine of the middle part, is equal to the rectangle under the tangents of the adjacent parts ; or', to the rectangle under the cosines of the opposite parts.