on AC, show that if in AC, CD be taken equal to CB, the rectangle AC . AD=CD2.

35. If c, c, be two chords of a circle that intersect at right angles, d its diameter, x, y the segments of the chord c, prove that 2(x2+y2)=d2 + c2-c12.

36. A circle is described about a triangle ABC, which has the side AB=side AC; from A a line is drawn meeting the base of the triangle in D and the circumference of the circle in E. Prove that the circle which passes through B, D, E touches AB.

37. ABCD is a parallelogram, and F a point in the diagonal AC; through F a line is drawn parallel to AB, meeting AD in G, and BC in H; and through F a line is drawn parallel to AD, meeting AB in E, and DC in K. Show that GK and EH produced meet in AC produced.

38. If two straight lines, AEB, CED, in a circle intersect in E, the angles subtended by AC and BD at the centre are together double of the angle AEC.

39. If a quadrilateral ABCD be inscribed in a circle,

AB.BC+AD.DC: BC. CD+BA. AD :: BD: AC.

40. Through a point K, within a parallelogram ABCD, straight lines are drawn parallel to the sides. Show that the difference of the parallelograms of which KA, KC are diagonals is twice the triangle BKD.

41. Show that the difference of the squares on the side of a regular pentagon and a regular decagon, inscribed in the same circle, is equal to the square on the radius.

42. Show that the equilateral triangle described about a circle has double the perimeter of that inscribed in the same circle.

43. If any number of triangles on the same base BC and on the same side of it have their vertical angles equal, and perpendiculars meeting in D be drawn from BC on the opposite sides, find the locus of D; and show that all the lines which bisect the angle BDC pass through the same point.

44. O is the middle point of the hypothenuse AB of a right-angled triangle ABC. Show that OA, OB, OC are all equal to one another.

45. Inscribe a sphere in a triangular pyramid.

46. Construct a square three times as large as a given square.

47. ACB is a triangle whose base AB is divided in E and produced to F, so that AE: EB:: AF: FB :: AC: CB. Prove that (EF-CF) (EF+CF)=EC2.

PLANE TRIGONOMETRY.

CXVI.

1. State the different methods of measuring angles. Show that English seconds may be reduced to French seconds by dividing by 324.

2. Define the sine, secant, and tangent of an angle. Show that the versed sine of an angle lies between 0 and 2. 3. Find cos 30° and sin 45°. Express all the trigonometrical ratios of an angle in terms of the versed sine.

4. Trace the changes of cot A in sign and magnitude, as A changes continuously from 0° to 360°.

5. Explain the terms complement and supplement, illustrating them by examples. Show, by example, how to convert English degrees into French. Explain the symbol and the No. 3·1416; and exhibit the algebraical signs of the sine, cosine, and tangent, in the third quadrant.

6. Show, geometrically, that sin (90°+A)=cos A; cos (90°+A)=—sin A; tan (—A)=—tan A.

7. Find the numerical values of sin 60°, cos 60°, tan 75°. 8. Prove that cos (A−B)=cos A. cos B+ sin A. sin B. Deduce the expression for sin (A-B).

If A+B is <90°, then cot A. cot B is >1.

9. Define the cosecant of an angle, and trace its variation in sign and magnitude through the four quadrants.

10. Prove

(1) sin A+ sin B=2 sin & (A+B) cos (A−B);

(2) tan (A-B)=

11. Show that

tan A-tan B

1+tan A tan B

tan (45° + A) ___2 cos A+ sin A+sin 3 A

tan (45°- A) 2 cos A-sin A-sin 3 A

12. Prove the formula

(1) 4 cos3 A sin 3 A+4 sin3 A cos 3 A=3 sin 4 A; (2) sin3 A sin 3 A+ cos3 A cos 3 A=cos3 2A.

cos A-cos 3 A

13. Prove that, (1)

=

tan 2 A;

sin 3 A-sin A

(2) (cos A+cos B)2 + (sin A+ sin B)2=4 cos2 (A-B);

1-tan A tan B

(3) cot (A+B) =

and hence find

tan A+tan B

cot 75°;

(4) 4 cos A cos (120°-A) cos (120°+A)=cos 3 A;

(2) cos A= {√1+sin 2 A+√1—sin 2 A}, A being

less than 45°.

(3) sin n A+sin (n−2) A=2 sin (n−1) A cos A.

15. Prove that

(1) (sin A+ sec A)2+(cos A+cosec A)2

=(1+sec A cosec A)2.

(2) sin 2 A sin A=cos A-cos A cos 2 A; and express cot 2 A in terms of cot A.

16. Trace the changes in the sign of sin A cos (A-15°) as the angle A increases from 0° to 360°.

17. Prove the formula

(1) cos (A+B)=cos A cos B-sin A sin B ;

sin A+ sin B

(2)

?=tan 1⁄2 (A+B) cot † (A−B).

sin A-sin B

18. Express sin A in terms of sin 2 A, and apply the result to find sin 105° from sin 210°——.

20. Prove that sin A={±√1+sin A±√I

cos A=}{÷|√1+sin A+√1—sin A}

Express the signs correctly when A lies between 315° and 360°; and find the sine of 9o.

21. (1) Prove cos 3 A=4 cos3 A-3 cos A. Find sin 18°, and thence deduce cos 36°.

(2) (cos A—cos B)2+(sin A—sin B)2=4 sin2 A—B,

(3) tan A+cot A=2 cosec 2 A.

(4) cos 10 A+ cos 8 A+3 cos 4 A+3 cos 2 A

=8 cos A cos3 3 A.

2

22. Obtain formulæ, adapted to logarithmic computation, for determining the angles of a triangle when its sides are known. If the sides be 2 ft., 8 ft., and 9.99992 ft., show that the cosine of the greatest angle is approximately, and 180π-1.8

that the angle contains approximately

degrees.

T.

3 tan A-tan3 A

23. Prove that tan A=

1-3 tan2 A

24. Without assuming the formula for cos (A+B), prove that cos 2 A=cos2 A-sin2 A.

M

27. What are the values of sin 30°, cos 30°, sin 45°, cos 45°? Deduce from them the value of sin 15° to 6 places of decimals.

28. In the triangle ABC, given a=3492-76, b=2471.34, B=20° 21', to find c.

29. If a, b, B are given to solve a triangle, b<a, and if c1, c2 be the two values found for determining c, the third side, prove that b2+c1 c2=a2.

30. Prove that (1) the sine of an angle, or of an arc, is equal to the cosine of its complement;

(2) cos A-cos B=-2 sin (A+B) sin † (A−B);

(6) tan A=2-√3, when sin 2A='5.

31. In a plane triangle c=732,b=846, a=945. Find the greatest angle.

and prove that sin (36°+A)—sin (36°— A)

sin A+ sin (72°+A)-sin (72°-A).

33. Investigate the area of a circle, and of any sector of a circle.

34. Prove that sin 4 A=4 sin A cos3A-4 cos A sin3A.