47. The area of a triangle is equal to half the product of two sides into the sine of the included angle.

Let ABC be any triangle, CD the perpendicular from C on the base AB.

48. To express the area of a triangle when one side and the angles are known.

With the figure of the preceding Article we have, as in Art. 39,

AC_AB sin ABC

sin ACB

therefore the area of the triangle

=

;

AB2. sin ABC. sin BAC

2 sin ACB

49. The area of a parallelogram is double that of a triangle having the same base and altitude: hence by Art. 47 the area of a parallelogram is equal to the product of two adjacent sides into the sine of the included angle.

Similarly we may apply Art. 48 to find the area of a parallelogram.

EXAMPLES. IV.

1. Given tan A=105, find the other Trigonometrical

8. Shew that the tangents of 60°, 45°, and 15o are in Arithmetical Progression.

9. At a distance of 100 feet from the foot of a tower the tower subtends an angle of 30°; find the height of the tower.

10. A base AB of 100 yards is measured close to the bank of a river, and a tree C on the other bank is observed from A and B; the angle CAB is found to be 60° and the angle CBA is found to be 45°: determine the breadth of the river.

11. A person standing on the bank of a river observes the angle subtended by a tree on the opposite bank to be 75°, and when he retires 20 feet from the river's edge he finds the angle to be 60°: determine the height of the tree and the breadth of the river.

12. Find the area of an equilateral triangle, each side being equal to a.

13. Find the area of an isosceles triangle, each of the equal sides being equal to a, and the included angle 30o.

14. A man 6 feet high standing at the top of a mast subtends an angle whose tangent is at a point on the deck 33 feet from the foot of the mast: find the height of the

mast.

15. The upper half of a post, seen from a point on a level with the foot of the post subtends an angle whose tangent is : find the tangent of the angle subtended by the whole post.

16. A staff at the top of a tower is observed to subtend an angle of 15o by an observer at a distance of a feet from the foot of the tower, and also to subtend the same angle when the observer is at a distance of b feet: find the height of the staff.

17. A column standing on a pedestal 25 feet 6 inches high subtends an angle of 45o at the eye of an observer who stands on the horizontal plane from which the pedestal springs. When the observer approaches 20 feet nearer to the column it again subtends an angle of 45° at his eye. Find the height of the column supposing the height of the observer's eye above the plane to be 5 feet 6 inches.

18. A person wishing to know the height of a wall, the foot of which was inaccessible, fixed an upright staff 5 feet high, (the height of his eye) at the place where the angular altitude above the level of his eye was 45°. Having then walked backwards till the angle between the top of the wall and the top of the staff was 18° 26', of which the tangent is, he found by actual measurement that his distance from the staff was 70 feet. Determine the height of the wall.

V. Logarithms.

50. The numerical calculations which occur in the solution of triangles are abbreviated by the aid of logarithms; and thus it is necessary to explain the nature and the properties of logarithms.

51. Suppose that a=n, then x is called the logarithm of n to the base a: thus the logarithm of a number to a given base is the index of the power to which the base must be raised to be equal to the number.

The logarithm of n to the base a is written log.n: thus if a*=n, then x=log. n.

52. For example 43=64, so that 3 is the logarithm of 64 to the base 4; or log, 64=3.

Again, required the logarithm of 27 to the base 9. Let x denote the required logarithm, so that 9*=27: thus (32)* = 33, that is 32=33; therefore 2x=3, that is x=1}.

In the next three Articles we shall give the properties on which the utility of logarithms chiefly depends.

53. The logarithm of a product is equal to the sum of the logarithms of its factors.

54. The logarithm of a quotient is equal to the logarithm of the dividend diminished by the logarithm of the divisor.

For let

therefore

xlog.m, and y=log.n;

m=a*, and n=a";

55. The logarithm of any power, integral or fractional, of a number is equal to the product of the logarithm of the number by the index of the power.

For let

therefore

m=a*; therefore m”=(a*)" =a**,

log. (m")=xr=rlog.m.

56. To find the relation between the logarithms of the same number to different bases.

Hence the logarithm of a number to the base b may be found by multiplying the logarithm of the number to the base a by log, a or by

1

log.b

we have

log, ax log,b=1.

57. There are two systems of logarithms which are used in Mathematics.

In one system the base is a certain number which cannot be expressed exactly; as far as nine places of decimals the number is 2.718281828.