By P. K. Jain, Ahmed Khalid
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Y/. y/: (22) Let us also determine the limit circle arc s in terms of the portion t of the tangent at the vertex of the axis drawn from an endpoint of the arc s, comprised between the contact point and the intersection point between the tangent and the axis drawn from the other endpoint of the arc s (Figure 15). t / t Figure 15. t /. In this ﬁgure are drawn the straight segment joining the endpoints of the limit circle arc, and a straight segment tangent to this limit circle arc. The vertical lines represent two axes through the endpoints of the limit circle arc.
B/ etc. A C C / D b sin A: The ﬁrst two of these equations are the well-known equations of ordinary trigonometry. The last two give ACB CC D : In order to give an example of the representation of curves by equations relating the coordinates of their points, let us denote by y the length of the perpendicular dropped from a point on a circle of radius r on a ﬁxed diameter of that circle, and x the portion of that diameter between the centre and the foot of the perpendicular y. x; y/-coordinates (Figure 11).
In this way, the aforementioned equations become sin A sin c D sin a; cos b sin A D cos B; cos a cos b D cos c: (5) Equations (5) concern a right spherical triangle that can be deduced from a right rectilinear triangle, and whose edges therefore cannot exceed 2 . Let us add that if we draw an arc of a great circle from the vertex of the angle A perpendicularly to the edge b, this arc will intersect the arc a or its extension in such a way that each of the arcs, from the intersection point until b, is equal to 2 , and the angle made by these arcs is b.
A Textbook Of Analytical Geometry Of Two Dimensions by P. K. Jain, Ahmed Khalid