Ex. 11.

10. If tan 0=tan3(1p), cos2p=÷(m2 — 1); find m in terms of ◊. 11. If tana tan3 (ß), tan ẞ=2 tan ; prove that 2p=a+ß. 12. If tan (45°-10)=tan3; prove that

ß=2

(tan + sec ) + (tan 0 — sec 0) = 2 cot 24.

13. If cos◊ = cos1a-sin2a(I —c2 sin20); prove that

1. Eliminate from the equations,

m=cosec―sin 0, n=sec-cos 0.

2. Eliminate from the equations,

(a+b) tan (0—4) = (a−b) tan (0+9), a cos 2p+b cos 20=c.

3. Eliminate and from the equations,

COS a

cos20=

cos2=

cos B
P

4. Eliminate and from the equations,

a sin2+b cos2=m, b sin2+a cos2=n, a tan 0=b tan 9.

5. Eliminate x, y, z from the equations,

b2 (x2 — y2) cos 0=a2z2 cos & ;

sin (0+)

=

sin (0-0)
y

sin 20

=

PROPERTIES OF PLANE FIGURES.

Ex. 13. In any right-angled triangle ABC, C being the right angle, and a, b, c the sides opposite to the angles A, B, C respectively; prove that—

8. R+r=(a+b); R, r being the radii of circles, one described about, the other inscribed in, the right-angled triangle.

Ex. 14. In any triangle ABC, having a, b, c sides opposite to the angles A, B, C respectively, prove that

b sin C

2. tan B=

tan B

a2+b2 — c2

3.

vers A

=

4.

a(a+c—b).

tan Ca2+ c—b2

2

5.

sin (A-B)_a2-b2
sin (A+B)

=

a+b

с

7. sin (AB)=.cos C.

=

vers B b(b+c-a)

6. cos A+ cos B= •(2sin2+C).

13. 1⁄2(a2+b2+c2)=ab cos C + ac cos B+bc cos A.

If R, r be the radii of circles, respectively described about, and inscribed in, any triangle; prove that

18. r=(a+b+c) tan A tan B tan C;

19. a cos A+b cos B+c cos C=4R sin A sin B sin C.

abc
a+b+c

20. If a, ß, y be the angles, which the sides of a triangle subtend at the centre of the inscribed circle; show that

Ex. 15.

4 sin a sin ẞ sin y=sin A+ sin B+ sin C.

1. If the sides a, b of a triangle include an angle of 120°, show that ca+ab+b2.

Ex. 15.

4. The vertical angle of an isosceles triangle is 90°; 2p is the perimeter of the triangle and r the radius of the inscribed circle; show that p: r=2+1: 2-1.

5. If a line CE bisect the angle C of any triangle and meet the

a+b
α- -b

base in E; tan AEC: ==

2ab tan C; CE= cos C.

a+b

6. Prove that the distance of the centre of the circle inscribed in

any triangle ABC from A is equal to

2bc
a+b+c

cos A.

7. Having given the perimeter 2p, and the three angles of a triangle; find the sides a, b, c.

8. If, in a triangle, the angles are such, that A: B: C=2:3:4;

9. If, in a triangle ABC, the angles are such, that A=2B=2°C; show that the side, a=2(a+b+c) sin 12°.

11. In any triangle, the length of a perpendicular from A on

b2 sin C+ c2 sin B

the opposite side= b+c

12. In any triangle the distance of a perpendicular drawn from

C on AB, from the middle point of AB=

13. If, in a triangle ABC, b-a-nc; show that

C

c tan A-tan B 2 tan A+tan B

cos A+
8(A + 2) =

B-A I+n cos B

=n cos -, and cot:

14. Given the three straight lines drawn from any point to the three angular points of an equilateral triangle; find one of its sides.

15. If through a point O within a triangle, three straight lines be drawn from the angles A, B, C, meeting the opposite sides

OD OE OF
AD BE+CF

=1.

16. If a straight line intersect the two sides AC, BC of a plane triangle in the points b, a, and the base AB produced in c; then Ab. Bc. Ca=Ac. Ba. Cb.

Ex. 15.

17. The sides of a triangle are 3, 5, 6; find the radii of the inscribed and circumscribed circles.

18. Two sides of a triangle are 3 and 12, and the contained angle is 30°; find the hypothenuse of an equivalent right-angled isosceles triangle.

19. In a triangle, having given B, a, and its area, construct the triangle.

20. Having given A, B, C the angles of a triangle, and R the radius of the circumscribing circle; find a, b, c.

21. If in a right-angled triangle, a perpendicular be drawn from the right angle to the hypothenuse; the areas of the two circles inscribed in the triangles on each side of this perpendicular are proportional to the corresponding segments of the hypothenuse.

22. If r be the radius of the circle inscribed in a triangle whose sides are a, b, c; and h, k, l be the distances of its centre from the hkl 2abc

angles of the triangle; show that r a+b+c

T3

23. If r,, r2, r, be the radii of circles which touch respectively a side of a triangle and the other two sides produced, prove that

r1r2r=abc cos A cos B cos C.

24. If r be the radius of the circle inscribed in a triangle, and r12, r, the radii of the circles inscribed between this circle and the sides containing the angles A, B, C respectively; prove that

(r1r2)2+(r1r;)2+(r2rz)±=r.

25. If r denote the radius of the circle inscribed in any triangle; r12, r, the radii of circles which touch each side respectively, and the other two produced; show that

I

I I I

Το ri

+

+; area of triangle=(r.r1r,r,).
73

26. The area of any triangle is to the area of the triangle whose sides are respectively equal to the lines joining its angular points with the middle points of the opposite sides as 4: 3.

27. The area of any triangle is to the area of the triangle formed by joining the points where the lines bisecting the angles meet the opposite sides as (a+b) (a+c) (b+c): 2abc.

23. The sides of a triangle are in Arithmetical progression, and its area is to the area of an equilateral triangle of the same perimeter as 35; find the ratio of the sides, and the value of the greatest angle.

29. Find the ratio between (1) the sides, (2) the areas, of an equilateral triangle and a square inscribed in the same circle.

Ex. 15.

30. Compare the areas (1) of regular pentagons, (2) of regular octagons described within and about a circle.

31. The square of a side of the regular pentagon inscribed in a circle is equal to the square of a side of the inscribed hexagon, together with the square of a side of the inscribed decagon.

32. If, in a regular polygon of n sides, each side is 2a; the sum of the radii of the circumscribed and inscribed circles is a cot

90°

n

33. The area of a regular polygon inscribed in a circle is a mean proportional between the areas of an inscribed and a circumscribed regular polygon of half the number of sides.

34. A, B, C, are 3 regular octagons; a side of A is equal to the diameter of the circle inscribed in B, or described about C; find the sum of the three areas, that of A being a2.

35. The distance between the centres of two wheels is a, and the sum of their radii is c; find the length of a string which crosses between them and just wraps round them.

36. If two circles, whose radii are a, b, touch one another externally, and if be the angle contained by the two common tangents to these circles; show that sin 0-4(a—b) √ ab

· (a+b)2

6. log sin 15°16'17"-9'420601; log cos 32° 14′ 55"=9'927237. 7. logtan 23°24′25′′=9.636370; log cot 53° 14′ 15′′=9·873364. 8. log sec 75°13′40′′=10′593499; log cosec 130° 29′ 30′′=10'1189c0.

I