Again, perimeter of the inscribed polygon = 2nr sin. n Area of inscribed polygon = n. A a Ob = n.‡a0.b0 sin a Ob 72. To find the area of a circle. This equals the area of the inscribed polygon when the number of sides is made infinite. be EXAMPLES. A. In the following examples the value of π is supposed to 22 7 1. Find the circular measures of 35o, 358. Ans. 6i, 55. 2. Find the degrees &c. in an angle whose circular measure is 1.21. Ans. 69°. 18'. 3. The radius of a circle is r; find the length of an arc which subtends 4° at the centre. TrA Ans. 180 4. The radius of the Earth being 4000 miles, what is the length of 1o of the meridian? Ans. 69 miles. 5. How far does a person at the equator travel in a second by reason of the Earth's rotation about its axis. Ans. 256 yards nearly. 6. The diameter of the Earth's orbit about the Sun being 192,000,000 miles, how far in space does the Earth travel in one second? Ans. 19 miles nearly. 7. The greatest square possible is cut out of a circle whose radius is r, the area of the remainder is (72) r2. 8. The area of a square inscribed in a circle: the area of an equilateral triangle inscribed in the same circle, as 8: 3/3. 9. If a1, a, a, be respectively the sides of a regular pentagon, hexagon, and decagon inscribed in a circle, then 1. Find the length of an arc subtending an angle of 60° in a circle whose radius is 105 feet. Ans. 110 feet. 2. If the length of an arc of 60° is 11 feet, then the radius of the circle is 10 feet 6 inches. 3. If a right angle were divided into 80 parts, and each of these into 80. Find the number of the latter parts contained in angle whose circular measure is '001. Ans. 45. 6. A circular flat object at distance a subtends an angle of 7. If a be the angle whose arc is r1, and radius the angle whose arc is r, and radius then 8. If R, r be radii of circles circumscribed about, and inscribed in a regular polygon whose side is 2a, then R-ra. 9. The Earth's radius being R and the height of a tower h; find how far an object at sea must be from the foot of the tower so as just to be visible from the top. 10. The height of one ship is n times that of another, and the top of one is just visible from the masthead of the other at the distance of a miles. Given R the Earth's radius, shew that the height of one is given by the equation 11. The radius of the circle of latitude 60o is 2000 miles. 12. What distance is travelled in an hour by a person situated in latitude 60o, by reason of the Earth's rotation? Ans. 261 miles. 13. If two plumb-lines suspended from points at a given distance apart on the Earth's surface are inclined at an angle of m"; and at n" when at an elevation h. Then the radius of 91 MISCELLANEOUS EXAMPLES. 1. Prove that the ratios of the angles denoted by a French and English degree, minute, second, are expressed by 3. Find the value of the interior angle of any regular polygon of n sides. n - 2 Ans. 180°. n 4. One regular figure has twice as many sides as another, and an angle of the first is one-third as large again as an angle of the second: find the interior angles of each. Ans. 144°, 108o. 5. The length of the line bisecting the angle a of a triangle and meeting the opposite side is 6. If regular pentagons be inscribed and circumscribed about a circle, then perimeter of the former the latter√5+1 4. 7. If cos 0 = cos a cos ẞ+ sin a sin ẞ cos A, and then sin' = sin ("+6+4) sin (+84). 2 perimeter of 11. If a cos +b cos (0+a) be put in the form A cos (0+ B) : find the values of A and B. Ans. A = √(a+b+2ab cos a), B=tan"1 b sin a a+b cos a 12. Divide a given angle a into two parts, whose sines are in the ratio m: n. Ans. sin1 m sin a sin-1 " n sin a √(m2+n2+2mn sin a) √(m2 + n2 +2 mn sin a) * 13. If cos (-a), cos &, cos (4+a) are in H. P. then 14. The elevation of a tower standing on a horizontal plane is observed; a feet nearer it is found to be 45°; b feet nearer still it is the complement of what it was at the first station; shew that the height of the tower is ab feet. a-b |