14. x sin - x2 sin 20 + x3 sin 30—4x+ sin 40+ &c. in infin.

15. (cos + sin ) + (cos + I sin 8)2 √ I √

16. (31—1)3 ̄1⁄2—†(32 − 1)3−2 +÷(31⁄2 − 1)3—§ — ... in inf.

17. + + + in inf.

2

I

2

32

I

...

18. + + +... in inf. 12 32 52

21. If tan (1+m sin 4) = m cos ; express in a series of sines and cosines of multiples of 4.

In any triangle ABC prove that

22. loge=(cos2B—cos2A) +÷(cos4B—cos4A) ++(cos6B—cos6A) + &c.

1. (a±b √=1)2=(a*+¿13) ~ (cos + sin), if tan

ᄒ

2. log, sec = tan20-tan40+ tan60-&c.; thence sum,

I

1−1+3−4+&c. in infinitum.

0 П+-9 2π+0

3. sin0=2"-1 sin - sin

-sin

sin

according as n is of the form 4m+1, or 4m+3 respectively.

5. sin sin 30 sin 50 . . . sin (2n + 1)0=2−";

9. Find the sum of the straight lines drawn, from any one of the angular points of a regular polygon of n sides, each side of which is 2a, to all the other angular points.

10. Find the product of all the lines that can be drawn from one of the angles of a regular polygon of n sides, inscribed in a circle of radius a to the other angles.

Ex. 22. Solve by Trigonometry the equations—

APPLICATION OF ALGEBRA TO GEOMETRY.

1. Divide a straight line, one foot long, in extreme and mean ratio.

2. In a right-angled triangle, the base-20, and the difference between the hypothenuse and perpendicular=8; determine the triangle.

3. Given the sum of the base and perpendicular of a right-angled triangle=49, and the sum of its base and hypothenuse=63; determine the sides.

4. The area of a right-angled triangle being 54, and the hypothenuse 15; determine the sides.

5. Determine the right-angled triangle, in which the hypothenuse is 17, and the radius of the inscribed circle 3.

6. The area of a right-angled triangle=840, and the radius of the circumscribed circle=29; determine the sides.

7. The perimeter of a right-angled triangle is 20, and the radius of the inscribed circle 14; find the sides.

8. The perimeter of a right-angled triangle is 24, and the perpendicular from the right angle on the hypothenuse is 4; find the sides.

9. Given the hypothenuse of a right-angled triangle, and the side of an inscribed square: find the two sides of the triangle (1) when the given side coincides with the hypothenuse, and (2) when an angle of the square coincides with the right angle of the triangle. 10. The area of a right-angled triangle being h, and the radius of the inscribed circle r; determine the sides.

11. CD is a perpendicular on the hypothenuse AB of a rightangled triangle; if r be the radius of the circle inscribed in ABC, and r,, r2 of those in CBD, ACD; show that r2=r‚2+r22.

12. If the base of a triangle be 6, and the two sides 3 and 4; find the segments of the base made by a line bisecting the vertical angle.

13. Two sides of a triangle are 5 and 6:4, and the length of a line bisecting the vertical angle and meeting the base is 4; find the base.

14. The base of a triangle is 14, the difference of the two sides 3, and a perpendicular from the vertical angle on the base 8; determine the sides.

15. The base of a triangle is 34, the sum of the two sides 50, and a perpendicular from the vertical angle upon the base 8; determine the sides.

16. Given the base of a triangle=50, the altitude=24, and the radius of the inscribed circle = 10; determine the sides.

17. Determine a triangle, having given 2d the sum of the two sides, p the perpendicular and 2n the difference of the segments of the base made by the perpendicular.

18. Given the base a, and the altitude p of a triangle; find the side of the inscribed square.

19. To find a triangle, such that its sides and a perpendicular on one of them from the opposite angle, may be in continued geometrical progression.

20. Given the three perpendiculars from the angles of a triangle upon the opposite sides; find the area and sides of the triangle.

21. Determine the sides of a triangle, which are in Ar. Prog. with the common difference=1, and in which the radius of the inscribed circle=4.

22. The sides of a triangle are in Ar. Prog., a, c being the longest and shortest sides: if R, r be the radii of the circumscribed and inscribed circles, show that 6Rr=ac.

23. Given a the base of a triangle, n: I the ratio of the two sides, and d the distance of the vertex from a given point in the base; determine the sides.

24. Given 2a the base, p the perpendicular and m2 the rectangle of the two sides of a triangle; find the sides.

25. From the obtuse angle A of a given triangle, draw to the base a line, the square on which shall be equal to the rectangle of the segments of the base.

26. Draw a straight line from one angle of a square whose side is 30, so that the part intercepted, between one of the sides containing the opposite angle and the other side produced, shall equal 16.

27. If an isosceles triangle be inscribed in a circle having each of the sides double of the base, show that 15(rad.)2= (2 side)2.

28. If a, b, c be the sides of a triangle, R, r the radii of the circumscribed and inscribed circles; show that

29. If in the triangle ABC, the lines bisecting the angles ABC and meeting the opposite sides a, b, c be h, k, l respectively; prove

30. Upon the sides of a triangle ABC right-angled at C, having described semicircles towards the same parts AEB, ADC, BDC; show that the difference of the figures ADBE, CD is equal to the triangle ABC.

31. The centres of three circles (A, B, C) are in the same straight line, B and C touch A internally, and each other externally; show that the portion of A's area which is outside B and C is equal to

the area of the semicircle described on the chord of A which touches B and C at their point of contact.

32. Through a point M equidistant from two straight lines AA', BB' at right angles to each other, to draw a straight line PMQ, so that the sum of the squares upon PM and MQ shall be equal to the square upon a given line b.

33. To inscribe a semicircle in a quadrant.

34. Find the side of a square, and the radius of a circle inscribed in a given quadrant.

35. To inscribe a circle in a given sector of a circle.

36. If a, b, c be the chords of three adjacent arcs of a circle whose sum equals the semicircumference, of which a is the radius; prove that 4x3- (a2 + b2+c2)x-abc=0.

37. Given the chords of two arcs of a given circle; find the chord of their sum, and the chord of their difference.

38. Given the lengths of two chords of a circle which intersect at right angles, and the distance of their point of intersection from the centre; find the diameter of the circle.

39. If a circle be described, so as to touch the side a of a triangle externally, and the sides b, c, produced; find its radius.

40. The radii of two circles which intersect one another are r, r'; and the distance of their centres is c; find the length of their common chord.

41. In a given square to inscribe another square having its side equal to a given straight line. What are the limits of this line?

42. If from one of the angles of a rectangle, a perpendicular be drawn to its diagonal d, and from the point of their intersection, lines p, q be drawn perpendicular to the sides which contain the opposite angle; show that, p3+q3= =d3.

43. If the distances of any given point within a square to three of its angular points be h, k, l; determine a side of the square. 44. If h, k,l be the sides of a regular pentagon, hexagon and decagon respectively, inscribed in the same circle; show that

45. The radius of a circle being 1, find the areas of the inscribed and circumscribed equilateral triangles; hence find the area of the inscribed regular hexagon.

46. Show that the area of a dodecagon inscribed in a circle is equal to that of a square on the side of an equilateral triangle inscribed in the same circle.

47. Find the radius of the circle inscribed in a given rhombus. 48. Show that, if in a quadrilateral figure the sums of the opposite sides are equal, a circle may be inscribed in it.

49. A circle is inscribed in an equilateral triangle, an equilateral triangle in the circle, a circle again in the latter triangle, and so on; if ",","2,3,... be the radii of the circles, prove that r=r,+r2+r+...