by substituting these values for sin B and cos B in the above equation, we have

By putting this value for sin B in the formula of Case 1, we have the following important formula for the area of a triangle:

S=√s(sa) (s—b) (s—c).

This is known as Heron's Formula for the area of a triangle, having been given in the works of this Greek writer. It is often given in geometry, but the proof by trigonometry is much simpler.

A special case of finding the area of a triangle when the three sides are given is that in which the radius of the circumscribed circle or the radius of the inscribed circle is also given.

If R denotes the radius of the circumscribed circle, we have, from § 106,

sin B

b

2 R

If denotes the radius of the inscribed circle, we may divide the triangle into three triangles by lines from the center of this circle to the vertices; then the altitude of each of the three triangles is equal to r. Therefore

S = {r(a + b + c) = rs.

By putting in this formula the value of S from Heron's Formula, we have

From this formula, r, as given in § 116, is seen to be equal to the radius of the inscribed circle.

Exercise 58. Area of a Triangle

Find the areas of the triangles in which it is given that:

52.3 rd.,

7. There is a triangular piece of land with sides 48.5 rd., and 61.4 rd. Find the area in square rods; in acres.

Find the areas of the triangles in which it is given that:

12. Given a = · 60, B = 40° 35′ 12′′, area = 12, find the radius of the inscribed circle.

Find the areas of the triangles in which it is given that:

13,

c = 37.

14. a =

408,

624,

b = 41,
b

C = 401.

205,

C= 445.

89

(=

23. If a is the side of an equilateral triangle, show that the area is a2 √3.

24. Two sides of a triangle are 12.38 ch. and 6.78 ch., and the included angle is 46° 24'. Find the area.

25. Two sides of a triangle are 18.37 ch. and 13.44 ch., and they form a right angle. Find the area.

26. Two angles of a triangle are 76° 54' and 57° 33′ 12", and the included side is 9 ch. Find the area.

27. The three sides of a triangle are 49 ch., 50.25 ch., and 25.69 ch. Find the area.

28. The three sides of a triangle are 10.64 ch., 12.28 ch., and 9 ch. Find the area.

29. The sides of a triangular field, of which the area is 14 A., are proportional to 3, 5, 7. Find the sides.

30. Two sides of a triangle are 19.74 ch. and 17.34 ch. The first bears N. 82° 30' W.; the second S. 24° 15' E. Find the area.

31. The base of an isosceles triangle is 20, and its area is 100 √3; find its angles.

32. Two sides and the included angle of a triangle are 2416 ft., 1712 ft., and 30°; and two sides and the included angle of another triangle are 1948 ft., 2848 ft., and 150°. Find the sum of their areas. 33. Two adjacent sides of a rectangle are 52.25 ch. and 38.24 ch. Find the area.

34. Two adjacent sides of a parallelogram are 59.8 ch. and 37.05 ch., and the included angle is 72° 10'. Find the area.

35. Two adjacent sides of a parallelogram are 15.36 ch. and 11.46 ch., and the included angle is 47° 30'. Find the area.

36. Show that the area of a quadrilateral is equal to one half the product of its diagonals into the sine of the included angle.

37. The diagonals of a quadrilateral are 34 ft. and 56 ft., intersecting at an angle of 67°. Find the area.

38. The diagonals of a quadrilateral are 75 ft. and 49 ft., intersecting at an angle of 42°. Find the area.

39. In the quadrilateral ABCD we have AB, 17.22 ch.; AD, 7.45 ch.; CD, 14.10 ch.; BC, 5.25 ch.; and the diagonal AC, 15.04 ch. Find the area.

40. Show that the area of a regular polygon of n sides, of which na2 180° one side is is a,

4

cot

n

41. One side of a regular pentagon is 25.

42. One side of a regular hexagon is 32.

43. One side of a regular decagon is 46.

Find the area.

Find the area.

Find the area.

44. If is the radius of a circle, show that the area of the regular

r

45. Obtain a formula for the area of a parallelogram in terms of

two adjacent sides and the included angle.

CHAPTER VIII

MISCELLANEOUS APPLICATIONS

119. Applications of the Right Triangle. Although the formulas for oblique triangles apply with equal force to right triangles, yet the formulas developed for the latter in Chapter IV are somewhat simpler and should be used when possible. It will be remembered that these formulas depend merely on the definitions of the functions.

Exercise 59. Right Triangles

1. If the sun's altitude is 30°, find the length of the longest shadow which can be cast on a horizontal plane by a stick 10 ft. in length.

2. A flagstaff 90 ft. high, on a horizontal plane, casts a shadow of 117 ft. Find the altitude of the sun.

10

30°

x

S

3. If the sun's altitude is 60°, what angle must a stick make with the horizon in order that its shadow in a horizontal plane may be the longest possible?

4. A tower 93.97 ft. high is situated on the bank of a river. The angle of depression of an object on the opposite

bank is 25° 12'. Find the breadth of the river.

5. The angle of elevation of the top of a tower

is 48° 19', and the distance of the base from the point of observation is 95 ft. Find the height of the tower and the distance of its top from the point of observation.

6. From a tower 58 ft. high the angles of depression of two objects situated in the same horizontal line with

the base of the tower, and on the same side, are

30° 13′ and 45° 46'. Find the distance between these two objects.

B

7. From one edge of a ditch 36 ft. wide the angle of elevation of the top of a wall on the opposite edge is 62° 39'. Find the length of a ladder that will just reach from the point of observation to the top of the wall.

8. The top of a flagstaff has been partly broken off and touches the ground at a distance of 15 ft. from the foot of the staff. If the length of the broken part is 39 ft., find the length of the whole staff.

9. From a balloon which is directly above one town the angle of depression of another town is observed to be 10° 14'. The towns being 8 mi. apart, find the height of the balloon.

10. A ladder 40 ft. long reaches a window 33 ft. high, on one side of a street. Being turned over upon its foot, the ladder reaches another window 21 ft. high, on the opposite side of the street. Find the width of the street.

11. From a mountain 1000 ft. high the angle of depression of a ship is 27° 35' 11". Find the distance of the ship from the summit of the mountain.

12. From the top of a mountain 3 mi. high the angle of depression of the most distant object which is visible on the earth's surface is found to be 2° 13' 50". Find the diameter of the earth.

13. A lighthouse 54 ft. high is situated on a rock. The angle of elevation of the top of the lighthouse, as observed from a ship, is 4° 52', and the angle of elevation of the top of the rock is 4° 2' Find the height of the rock and its distance from the ship.

14. The latitude of Cambridge, Massachusetts, is 42° 22' 49". What is the length of the radius of that parallel of latitude?

15. At what latitude is the circumference of the parallel of latitude equal to half the equator?

16. In a circle with a radius of 6.7 is inscribed a regular polygon of thirteen sides. Find the length of one of its sides.

17. A regular heptagon, one side of which is 5.73, is inscribed in a circle. Find the radius of the circle.

18. When the moon is setting at any place, the angle at the moon subtended by the earth's radius passing through that place is 57' 3". If the earth's radius is 3956.2 mi., what is the moon's distance from the earth's center?

19. A man in a balloon observes the angle of depression of an object on the ground, bearing south, to be 35° 30'; the balloon drifts. 23 mi. east at the same height, when the angle of depression of the same object is 23° 14'. Find the height of the balloon.

20. The angle at the earth's center subtended by the sun's radius is 16' 2", and the sun's distance is 92,400,000 mi. Find the sun's diameter in miles.