DESCRIPTIVE GEOMETRY. ORTHOGRAPHIC AND HORIZONTAL PROJECTIONS. Define the terms "line of level," "planes of projection," "trace," and "plan." 686. Show that the straight line which joins the projections of the same point is perpendicular to the line of level. 687. If a plane be perpendicular to one of the planes of projection, prove that its trace upon the other plane is perpendicular to the line of level. 688. Prove that the horizontal and vertical projections of a point are in the line drawn through either of them, perpendicular to the line of level. 689. State the principles upon which the projections of the intersection of two planes whose horizontal and vertical traces are given, may be determined. 690. Prove that the horizontal trace of the vertical projecting plane of a straight line is perpendicular to the line of level. 691. The plan and elevation of a straight line being given, find its traces. 692. The traces of a straight line being given, determine its plan and elevation. 693. Find the intersection of a given line with a given plane. 694. Find the point of intersection of three given planes. 695. The plan of a straight line in a given plane being given, find the elevation of that line. 696. Through a given point draw a plane parallel to two given straight lines. 697. From a given point draw a straight line perpendicular to a given plane, and find its true length. 698. Construct the angle of inclination of two given planes. 699. The true representation of any plane rectilineal figure upon a given plane, being given, find the plan and elevation of the figure. 16 700. If (a,b) be the horizontal projection of a straight line; state clearly the principles on which you would determine its absolute length, and its inclination to the horizon. 701. The horizontal projections and indices of two points being given, state how the inclination to the horizontal plane of the straight line joining these points may be determined. 15 702. If the horizontal projections of the points a, b, be twenty units of the scale of the plan distant from one another, at what distance from the former will be the horizontal trace of the line joining those points? 703. The scales of slope of two planes being given, determine the projection of the line of intersection of those planes. 704. The scale of slope of a plane being given; show how its trace upon the horizontal plane may be found. 705. Show that in a system of horizontal contours the increasing proximity of the lines indicates an increasing degree of slope; and vice versa. PLANE TRIGONOMETRY. 706. Define "the chord," "the tangent," and "the secant" of an arc; and show that chd 2a = 2 sin a. 707. Express the "tangent" and "secant" of an arc in terms of its "sine." 708. Show that if a° be the angle in a segment of a circle whose radius is r, the base of the segment is 2r sin ao. 709. The difference of two arcs of a circle is 20 grades, and their sum is 48 degrees; find the arcs. 710. Find, geometrically, the values of tan 30°, tan 45o, and tan 60o. 711. Find the angle whose circular measure is 3. 713. Give accurate line-definitions of the sine, tangent, secant, and cosine of an arc of a circle; and determine the numerical values of cos 30°, sin 45°, sec 45°, and tan 120°; the radius of the circle being unity in the first two examples, and 10 in the latter two. 714. Trace by a figure the changes in the cosine of an arc of a circle, as the arc varies from 0 to 360°. 715. The radius of a circle being 16 feet, find the number of degrees, &c. in an arc of 6 feet, and also the circular measure of the subtending angle at the centre. 716. Find the values of tan 30°, tan 60°, tan 45° and tan 7°30′, and thence determine tan 15° and tan 75o. 717. Find the length of the arc whose circular measure is 2. 719. In a right-angled triangle ABC, express the ratios as trigonometrical functions of the angle A, when C is the right angle. 720. The length of an arc in inches is expressed numerically by five times its circular measure, and by ten times the reciprocal of its angular measure in degrees; find the length of the radius, and the number of degrees in the arc. Investigate the following relations : 725. sin A+ sin B = 2. sin (A + B) cos § (A – B). 732. An arc of 40° on a circle whose radius is 6 inches is found to measure a inches; and this length applied to the cir cumference of another circle is found to cover an arc of 25°; what is the radius of the latter circle? 733. Give the algebraic signs of the sine, cosine, tangent and secant of each of the arcs 70°, 110°, 195°, and 280°; illustrating the result given by a geometrical figure. 734. Prove that sin 30° and sin 45° = cos 45° = 1 2' and radius unity; and thence find the values of tan 30°, sin 75o, 744. 1 + cos 20. cos 24 = 2 (sin2 0 . sin2 + cos 0. cos2 ). |