n 200 14. If n be any whole number, shew that of a right angle contains a whole number both of English minutes and of French minutes. 15. What must be the unit angle if the sum of the measures of a degree and a grade is I? 16. An angle is made up of two parts, one containing a degrees and the other containing b grades: compare the angle with a right angle. 17. If A+B+C=180°, shew that tan A+tan B+tan C=tan A tan B tan C. 18. If the tangents of the three angles of a triangle be as the numbers 1, 2, 3, shew that they must be equal to 1, 2, 3. 19. Having given tan (a+0)tan (a− 0)=k, find sin 0. 21. If cos (s-2a) + cos (s − 2ẞ) = cos (s — 2y) + cos (s — 28), where s=a+B+y+d, then tan a tan ẞ=tan y tan d. 22. Shew that 2 cos 5° 37′ 30′′ = √[2+ √{2+√(2+ √√2)}]. is either isosceles, or such that the sides are in Arithmeti cal Progression. XXIII. Area of a Circle. 217. The principal object of the present Chapter is to find an expression for the area of a circle; we shall give some important preliminary propositions. 218. If be the circular measure of a positive angle less than a right angle, ✪ is greater than sin 0 and less than tan 0. Let AOB be an angle less than a right angle, and let OB=0A; from B draw BM perpendicular to OA and produce it to C.so that MC=MB; draw BT at right angles to OB meeting OA produced at T, and join CT. Then the triangles MOC and MOB are equal in all respects, so that the angle TOC=the angle TOB; therefore the triangles TOC and TOB are equal in all respects, so that TCO is a right angle, and TC=TB. With centre O and radius OB describe an arc of a circle BAC; this will touch BT at B and CT at C. Now we assume as an axiom that the straight line BC is less than the arc BAC; thus BM, the half of BC, is less BM than BA, the half of the arc BAC; therefore is less BA than ; that is, the sine of AOB is less than the circular OB measure of AOB. Again, we assume as an axiom that the arc BAC is less than the sum of the two exterior straight lines BT and BA TC; thus BA is less than BT; therefore is less than OB BT OB; that is, the circular measure of AOB is less than the tangent of AOB. Hence sin 0, 0, and tan 6 are in ascending order of magnitude if o be less than 219. The limit of minished is unity. π 2 For sin 0, 0, and tan 6 are in ascending order; divide ; but when is zero, cos e is unity; hence as 220. To find the area of a circle. Let r be the radius of a circle. Suppose a regular polygon of n sides described about the circle. Then the circular measure of the angle which each side subtends at 2π n the centre of the circle is ; and therefore, by Art. 139, Suppose n to increase without limit, then the area of the polygon approximates continually to the area of the circle as the limit, and therefore the area of the circle will be equal to the limit of the above expression. But when n is indefinitely great therefore the area of a circle of radius r=πr2. 221. To find the area of a sector of a circle. Let be the circular measure of the angle of the sector; then, by Euclid vi. 33, Thus the area of a sector is equal to half the product of the square of the radius into the circular measure of the angle. Since 0 is the circular measure of the angle the length of the arc of the sector is re; hence the area of a sector is equal to half the product of the length of the arc into the radius. EXAMPLES. XXIII. 1. With the angular points of an equilateral triangle as centres, and a radius equal to half the side, arcs are described touching each other: determine the area of the figure which they form. 2. A chord of length r is placed in a circle of radius r: determine the areas of the two segments into which the chord divides the circle. 3. A circle is described round a triangle the angles of which are 45°, 60°, and 75° respectively determine the areas of the segments of the circle cut off by the sides. 4. Two circles touch each other, and a common tangent is drawn. Supposing their radii to be r and 3r respectively, shew that the area of the curvilinear triangle bounded by the two circles and the common tangent is value, shew that 0 is greater than √3−1. 7. Shew by Arts. 104 and 188 that in any triangle |