TRIGONOMETRY. PLAIN Trigonometry, is the art of measuring plain triangles, by comparing sides and angles together, by known analogies; whereby three things being given, a fourth may be found, on condition that one of them be a side: to do which, right lines are applied to the arch of a circle, described on the angular point, viz. A chord is a line that divides the circle into two unequal parts, and is a chord to them both, as DH is the chord of the arches DH and DAH. 2. The sine of an arch, is a line drawn from one end, or termination of the arch, perpendicular to the radius, or it is half the chord of twice the arch, so that RS is the sine of the arch AS, and SZ the co-sine: the sine and co-sine making a quadrant, or 90°. 3. A versed sine, is that part of the diameter contained between the sine, and the arch, as RA and RCD, and is the versed sine of SHD, or DEP its equal. 4. A tangent of an arch, is a line drawn perpendicular to one end of the diameter, just touching the arch, as AT is the tangent of the arch AS, and HG the co-tangent: the tangent and co-tangent making á quadrant, or 90°. 5. A secant of an arch, is a line drawn from the centre through the circumference, until it cuts the tangent, as CT is a secant of the arch SA, and GC the co-secant, meeting the co-tangent: the secant and co-secant also making a quadrant, or 90°. RIGHT ANGLED TRIGONOMETRY. THE solution of the several cases in Right Angled Trigonometry, depends on the following Position, which ought to be well committed to memory, and, by comparing it with the annexed figures, clearly understood by the learner, before he proceeds. POSITION. In every right angled triangle, If the hypothenuse be made the radius of a circle, the other two sides, or legs, will be, each the sine of its opposite angle. If either leg, including the right angle, be made the radius of a circle, the other leg will be the tangent of its opposite angle, and the hypothenuse the secant of the same angle. The foregoing Position, accurately compared with the following figures, will be more instructive, than lengthy demonstrations. K Note. When the hypothenuse is made radius, then the base is the sine of the opposite angle C ; and the perpendicular, a sine of the opposite angle A. When the perpendicular is made radius, then the base is tangent of the opposite angle C, and the hypothenuse a secant of the same angle. And when the base is made radius, then the perpendicular is tangent of the opposite angle A, and the hypothenuse a secant of the same angle. When the angles, and one side are given, to find either, or both the other sides; then either side may be madé radius; and each, in rotation, to prove the work. RULE. As the name on the given side, Is to the same side; So is the name on the side required, |