Trigonometry, Plane and Spherical: With the Construction and Application of LogarithmsKimber and Conrad, 1810 - 125 páginas |
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Página 9
... difference of the sides to twice the distance DE of the perpendicular from the middle of the base . From the vertex ... half the sum of the two opposite angles , to the tangent of half their difference . For , let ABC ( fig . 5. ) be the ...
... difference of the sides to twice the distance DE of the perpendicular from the middle of the base . From the vertex ... half the sum of the two opposite angles , to the tangent of half their difference . For , let ABC ( fig . 5. ) be the ...
Página 10
... equal to half the sum of the angles opposite to the sides proposed . More- over , since ABC = ABD ( ADB ) + DBC , and C = ADB -DBC ( by 32. 1. ) it is plain that ABCC is = 2DBC ; or that DBC is equal to half the difference of the same ...
... equal to half the sum of the angles opposite to the sides proposed . More- over , since ABC = ABD ( ADB ) + DBC , and C = ADB -DBC ( by 32. 1. ) it is plain that ABCC is = 2DBC ; or that DBC is equal to half the difference of the same ...
Página 11
... half the sum of the required angles to the tangent of half their difference * . * This theorem , though it requires two proportions , is commonly used by astronomers in determining the elongation and parallaxes of the planets ( being ...
... half the sum of the required angles to the tangent of half their difference * . * This theorem , though it requires two proportions , is commonly used by astronomers in determining the elongation and parallaxes of the planets ( being ...
Página 16
... difference BC or CD , so is the sine CF of the mean , to half the sum of the sines BE + DG , of the two extremes : and , as radius to the sine of the common difference , so is the co - sine FO of the mean to half the difference of the ...
... difference BC or CD , so is the sine CF of the mean , to half the sum of the sines BE + DG , of the two extremes : and , as radius to the sine of the common difference , so is the co - sine FO of the mean to half the difference of the ...
Página 20
... ( equal to half the side of a decagon inscribed in the circle ) , let the co - sine of 3 ° , the difference * The side of a hexagon being equal to the radius of its circumscribing circle ( 15. 4. ) , it follows that the sine of 30 ...
... ( equal to half the side of a decagon inscribed in the circle ) , let the co - sine of 3 ° , the difference * The side of a hexagon being equal to the radius of its circumscribing circle ( 15. 4. ) , it follows that the sine of 30 ...
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Términos y frases comunes
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
Pasajes populares
Página 69 - TO THEIR DIFFERENCE ; So IS THE TANGENT OF HALF THE SUM OF THE OPPOSITE ANGLES', To THE TANGENT OF HALF THEIR DIFFERENCE.
Página 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...
Página 25 - The cotangent of half the sum of the angles at the base, Is to the tangent of half their difference...
Página 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.
Página 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...
Página 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.