[In what follows, S denotes the area of a triangle, s its semi-perimeter, R, r, ra, rb, rc, the radii of its circumscribing, inscribed, and escribed circles, respectively.] 1. Prove that any side of a triangle is equal to the second side into the cosine of the angle opposite the third sine plus the third side into the cosine of the angle opposite the second side. 2. Derive expressions, in terms of the sides of a given triangle, for the radii of its circumscribing circle, and of the four circles which touch the sides. 3. Derive expressions for the radii in Ex. 2, in terms of the sides and the area of the given triangle. similar formulas for R and r. Write similar formulas for rьrc. 10. Show that the length of the tangents to the inscribed circle from the angle A is s - a, from the angle B is s b, from the angle C is s - C. 11. Write and derive the formula for the area of a triangle: (a) in terms of the three sides; (b) in terms of two sides and their included angle; (c) in terms of one side and the two angles adjacent to it. 12. (a) Prove that S = s(s c) when C = 90°; (b) if x, y, be the lengths of the two diagonals of a parallelogram, and the angle between them, show that areaxy sin 0. 13. Find the areas of some of the triangles in Ex. 10, Chap. VII Find the radii of their circumscribing, inscribed, and escribed circles. 14. (a) An isosceles triangle whose vertical angle is 78° contains 400 square yards; find the lengths of the sides. (b) Find two triangles each of which has sides 63 and 55 ft. long, and an area of 874 sq. ft. (c) The angles at the base of a triangle are 22° 30′ and 112° 30′ respectively; show that the area of the triangle is equal to the square of half the base. 15. (a) Show that the area of a regular polygon inscribed in a circle is a mean proportional between the areas of an inscribed and circumscribing polygon of half the number of sides. (b) The sides of a triangle are as 2:3:4; show that the radii of the escribed circles are as : } : 1. 16. Two roads form an angle of 27° 10' 25". At what distance from their intersection must a fence at right angles to one of them be placed so as to enclose an acre of land? 17. If the altitude of an isosceles triangle is equal to its base, the radius of the circumscribing circle is of the base. 18. An equilateral triangle and a regular hexagon have the same perimeter. Show that the areas of their inscribed circles are as 4:9. 19. If the sides of a triangle are 51, 68, and 85 ft., show that the shortest side is divided by the point of contact of the inscribed circle into two segments, one of which is double the other. CHAPTER IX. 1. Explain how angles are measured (1) by sexagesimal measure, (2) by radian measure. Show how to connect the radian measure of an angle with its measure in degrees. Find the number of degrees in the angle called the radian. How many degrees are there in an arc whose length is equal to the diameter? Show that the radian measure of an angle is the ratio of the lengths of two lines. What advantage is there in using radian measure? 2. (a) Give the number of degrees in each of the following angles: 1TM, π, 2 π, ¡π, n«,, − 3 π, 3 π, 3 π, § π, } π, Jπ, 8×, −1, −1π, − Zir, - § T, π 3' π ηπ, ᅲ, (§)(~), (2})(~), (− 4)(r). (b) Give the supplements and complements of those angles in radian measure and in degree measure. (c) Give the radian measures of 30°, 80°, 49°, 41° 30′ 15′′, 120°, — 210°, – 175°. Give the radian measures of their supplements and complements. 3. (a) A central angle 1.25" is subtended by a circular arc of 16 ft.; find the radius. (b) Find the number of radians and degrees in the central angle subtended by an arc 9 in. long, in a circle whose radius is 10 ft. QUESTIONS AND EXERCISES. 195 (c) Find the radius of a circle in which an arc 15 in. long subtends at the centre an angle containing 71° 36′ 3.6. (d) If the radius be 8 in., find the central angle, in degrees and in radians, that is subtended by an arc 15 in. long. (e) An angle of 3r is subtended by an arc of 5 in.; find the length of the radius; find also the number of radians, and of degrees, in an arc of 1.5 in. (ƒ) Find the number of radians and seconds in the angle subtended at the centre of a circle whose radius is 2 mi., by an arc 11 in. long. (g) Find the length of the arc which subtends a central angle of (1) 2 radians, the radius being 10 in.; (2) 1.5 radians, radius 2 ft.; (3) 4.3 radians, radius 21 yd.; (4) 1.25 radians, radius 8 in. 4. The value of the division on the outer rim of a graduated circle is 5', and the distance between the two successive divisions is .1 of an inch. Find the radius of the circle. 5. Show that the distance in miles between two places on the equator, which differ in longitude by 3° 9', assuming the earth's equatorial diameter to be 7925.6 mi., is 217.954 mi. 6. (a) The difference of two angles is 10°, the radian measure of their sum is 2. Find the radian measure of each angle. (b) One angle of a triangle is π degrees, another is π grades. Show that the radian measure of the third angle is π 19T2 1800 (c) If the number of degrees in an angle be equal 5π 19 to the number of grades in the complement of the same angle, prove that the radian measure of the angle is ratios 1:2:3. Express their magnitudes in each of the three systems of angular measurement. (e) One angle of a triangle is 45°, another is 1.5 radians. Find the third, both in degrees and in radians. (f) Express in degrees and in radian measure the vertical angle of an isosceles triangle which is half of each of the angles at the base. (d) The angles of a triangle are in the 7. Prove the following statements, in which a denotes the length of a side of a regular polygon; P, the length of its perimeter; n, the number of its sides; r, the radius of the inscribed circle; R, the radius of the circumscribing circle: 1 Define and illustrate the trigonometric functions. Show in tabular form the signs of these functions in each of the four quadrants. 2. (a) Construct a table showing the values, with proper signs, of the trigonometric functions of 0°, 30°, 45°, 60°, 120°, 180°, 225°, 270°, 315o, 360°. (b) Compare the trigonometric functions of 90° — A, 90° + A, 180° + A, 180° - A, A, with those of A. 3. Show, from both the ratio and the line definitions of the trigonometric functions, that (1) the sine and cosine are never greater than unity, (2) the cosecant and secant are never less than unity, (3) the tangent and cotangent may have any values whatever from negative infinity to positive infinity, (4) the trigonometric functions change signs in passing through zero or infinity, and through no other values. 4. Given tan A tions of A. = -1, find the values of the other trigonometric func 5. Find geometrically an expression for the cosine of the difference of two angles in terms of the trigonometric functions of those angles. 6. Prove that: (a) sin2 B + sin2 (A − B) + 2 sin B sin (A – B) cos A cos (n + 2) A sin (n + 2) A sin nA tan (n + 1)A. 7. Give the ratio definitions of the trigonometric functions, sine, cosine, tangent, and secant. These functions have also been defined as straight lines. Give these definitions, and show from them that tan 90°, sec 90° would each be infinite. Show that the two systems are consistent. 8. (a) Trace the changes, in magnitude and sign, in the values of the trigonometric functions as the angle increases from 0° to 360°. (b) Trace the changes of sign of sine as increases through 360°, and show that its 0 0 2 equivalent 2 sin COS has always the same sign as sin 0. 9. Trace the changes, as A increases from 0° to 180°, in the sign and value of (a) cos (π sin A), (b) sin A+ cos A, (c) sin A cos A. Draw the graphs of these functions. 10. Show that the radian measure of an acute angle is intermediate in value between the sine and the tangent of the angle. 11. (a) Show that the limit of when is indefinitely diminished is 1, sin 0 tan 0 i.e. sin = 9, very nearly. (b) Show that the limit of when is indefi 12. Find the area of a regular polygon of n sides inscribed in a circle, and show, by increasing the number of sides of the polygon without limit, how the expression for the area of the circle may be obtained. 13. (a) Find the distance at which a building 50 ft. wide will subtend an angle of 3'. (b) A church spire 45 ft. high subtends an angle of 9' at the eye. Find its distance approximately. (c) Find approximately the distance of a tower 51 ft. high which subtends at the eye an angle of 55'. (d) How large a mark on a target 1000 yd. off will subtend an angle of 1" at the eye? QUESTIONS AND EXERCISES. 197 14. Show how the functions may be represented by lines connected with a circle. 15. Explain, with illustrations, how functions may be graphically represented by means of a curve. Draw the graphs of the trigonometric functions. NOTE. "If a function of a variable has its magnitude unaltered when the sign of the variable is changed, that function is called an even function, but if the function has the same numerical value as before, but with opposite sign, then that function is called an odd function; for instance, x2 is an even function of x, x3 is an odd function of x, but x2 + x3 is neither even nor odd, since its numerical value changes when the sign of x is changed." 16. Show that the cosine, secant, and versine of an angle are even functions, and the sine, tangent, cotangent, and cosecant are odd functions, and the coversine is neither even nor odd. (See Art. 78.) CHAPTER XI. N.B. The problems which are purely numerical are to be solved independently of tables. The results can be verified by means of the tables. 1. (a) Deduce a general expression for all angles which have the same sine; (b) for all which have the same cosine; (c) for all which have the same tangent. (d) What are the general expressions for all angles which have the same secant, cosecant, cotangent, respectively? 2. Define inverse trigonometric functions; give illustrations. Define tan-1x, cos-1x. 3. (a) Explain fully the equations sin (sin-1 })=}, sin-1(sin 0)=0. (b) Construct sin-1 (), cos-10, tan-1 ∞o, sec-1 sec -1sec 2). (c) Find tan (cos−1 §). [Carefully state the limitations under which the following equations are true.] 4. Show that: (a) tan-1x+tan-1y=tan−1 x+Y; (b) tan-1x-tan-1y= tan-1 1-xy xy; (c) sin-1x — sin-1y: =cos-1√1 - x2 — y2 + x2y2 + xy. 1 + xy 5. (a) From sin 2 A=2 sin A cos A, show that 2 sin-1 x=sin-1(2 x√1−x2). (b) Show that for certain values of the angles, 2 cos-1 x = cos ̄1 (2 x2 - 1); 2 tan-1x tan-1 2x 2 cot-1 x = cosec-1 1. 1 + x2 1-x 6. Show that for certain values of the angles: (a) cos-1x=sin-1- + 2 (c) tan-1m + York Pubs Libr. CHATHAM S 23 EAST EN |