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Side 39 - 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.
Side 32 - The circumference of every circle is supposed to be divided into 360 equal parts, called degrees ; each degree into 60 equal parts, called minutes ; and each minute into 60 equal parts, called seconds.
Side 78 - Required the height and distance of the steeple. Ans. Height, 210.4 feet; distance, 250.8 feet. 14. Two pulleys, whose diameters are 6 inches and 4 feet 3 inches, respectively, are placed at a distance of 3 feet 6 inches from centre to centre. What must be the length of a belt which shall connect them, by passing around their circumferences, without crossing...
Side 78 - From the top of a tower, whose height is 108 feet, the angles of depression of the top and bottom of a vertical column standing in the horizontal plane are found to be 30° and 60° respectively ; required the height of the column.
Side 80 - Ans. 41.9968. 6. tin a level garden there are two lofty firs, having their tops ornamented with gilt balls, one is 100 feet high, the other 80, and they are 120 feet distant at the bottom ; now the owner wants to place a fountain in a right line between the trees, to be equally distant from the top of each...
Side 95 - TO THEIR DIFFERENCE ; So IS THE TANGENT OF HALF THE SUM OF THE OPPOSITE ANGLES', To THE TANGENT OF HALF THEIR DIFFERENCE.
Side 98 - 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.
Side 36 - The side of a regular hexagon inscribed in a circle is equal to the radius of the circle.
Side 117 - The straight line joining the vertex and the centre of the base is called the axis of the cone.
Side 40 - Def. 10. 1.) If then CE is made radius, GE is the tangent of GCE, (Art. 84.) that is, the tangent of half the sum of the angles opposite to AB and AC. If from the greater of the two angles ACB and ABC, there be taken ACD their half sum ; the remaining angle ECB will be their half difference.