In general, if we represent any angle by A, Cos. A sine (90°-A). Cot. A tang. (90°-A). Cosec. A sec. (90°-A). Since, in a right-angled triangle either of the acute angles is the complement of the other, the sine, tangent, and secant of one of these angles is the cosine, cotangent, and cosecant of the other. The sine, tangent, and secant of an arc are equal to the sine, tangent, and secant of its supplement. Thus FG is the sine of the arc AF, or its supplement, BDF. Also, AI, the tangent of the arc AF, is equal to BM, the tangent of the are BDF; and CI, the secant of the arc AF, is equal to CM, the secant of the arc BDF. The versed sine of an acute angle, A CF, is equal to the radius minus the cosine CG. The versed sine of an obtuse angle BCF, is equal to radius plus the cosine CG; that is to BG. The relations of the sine, cosine, etc., to each other, may be derived from the proportions of the sides of similar triangles. Thus the triangles CGF, CAI, CDL, being similar, we have, 1. CG :: : CF : CA AI; that is, representing the arc by A, and the radius of the circle by R, cos. A sin. A: R : tang. A; : : 4. GF : CF :: CD : CL; that is, sin. A : R :: R cosec A; 5. AI : AC :: CD : DL; that is, tang. A : R :: R cot. A. Also in the right-angled triangle CGF, we find CG2 GF2=== + CF2; that is, sin. 2A + cos. 2A— R2; or, the square of the sine of an arc, together with the square of its cosine, is equal to the square of the radius; 1. That a straight line may be drawn from any one point to any other point. 2. That a terminated straight line may be produced to any length in a straight line. 3. And that a circle may be described from any center, and with any radius. 4. That no part of the circumference of a circle is straight. 5. That it is possible to find a straight line equal in length to the circumference of a given circle. 6. That it is possible to find a common measure between the side and diagonal of a square in integers to infinity. 7. That it is possible to inscribe in any circle a polygon with any given number of sides expressed by 2+1, provided, 2 + 1 is a prime number. AXIOMS. n 1. Things which are equal to the same thing are equal to one another. 2. If equals be added to equals the wholes will be equal. 5. If equals be taken from unequals the remainders will be unequal. 6. Things which are double of the same thing are equal to one another. 7. Things which are halves of the same thing are equal to one another. 8. Magnitudes which coincide with one another, that is, which exactly fill the same space are equal to one another. 9. The whole is greater than its part. 10. Two straight lines can not inclose a space. 11. All right angles are equal to one another. 12. If a straight line meet two straight lines, so as to make the two interior angles on the same side of it taken together less than two right angles, these straight lines being continually produced, shall at length meet on that side on which are the angles which are less than two right angles. ARTICLES. 1. The double of the cosine or secant is the constantly assumed diameter. 2. The sum of the sines or tangents is the constantly assumed circumference. 3. The sine is a mean proportion between the double of the cosine and the second tangent. 4. The second tangent is a mean proportional between the double of the second secant and the third tangent, etc., etc., etc. 5. The rectangle contained by the sine and cosine is equal to the area of the inscribed double triangle. 6. The rectangle contained by the radius and the tangent is equal to the area of the circumscribed double triangle. 7. The rectangle contained by the radius and the sine is a mean proportional between the inscribed and circumscribed double triangles of half the number of sides. 8. The rectangle contained by the inscribed double triangle and the number of sides of the entire polygon is equal to the area of the inscribed polygon. 9. The rectangle contained by the circumscribed double triangle and the number of sides of the entire polygon is equal to the area of the circumscribed polygon. 10. The rectangle contained by the rectangle of the radius and the sine and the number of sides contained in the given polygon, is equal to the area of the entire inscribed polygon of double the number of sides. SIGNS. The following are the principal signs employed: The Sign of Addition, +, called plus: Thus, AB, indicates that B is to be added to A. The Sign af Subtraction, called minus: Thus, A B, indicates that B is to be substracted from A. |