If now the line OP be pictured as revolving from the position OL, the sine of the angle XOP, namely MP, will be seen to increase from zero and approach unity as the angle approaches 90°. The cosine, namely OM, decreases from unity to zero, and the tangent (LQ) increases without limit. Also, as the angle increases beyond 90°, the directions of the lines MP and OM indicate the signs of the sine and cosine. The other functions follow directly from these two by virtue of the relations of Art. 12.

CHAPTER III

THE SOLUTION OF RIGHT TRIANGLES. LOGARITHMS AND COMPUTATION BY MEANS OF LOGARITHMS

28. Solution of Right Triangles. With the definitions of the trigonometric functions and tables giving their numerical values we are now prepared to solve right triangles; that is, to find the values of the unknown parts from those that are known.

K

A=32°16′

b

FIG. 17.

B

a=124

C=90°

Two parts in addition to the right angle must be known, and one at least of these parts must be a side. We have then the general rule of procedure: Select that trigonometric function which involves the two known parts and one unknown part. The value of the un

known part can then be computed by elementary algebraic

processes.

Example 1. Given A = 32° 16′, a 32° 16′, a = 124, C′ = 90°, find B, b, and c. See Fig. 17.

or

Obviously B 90° - A 90° 32° 16' 57° 44'. Then = — —

=

From the tables we find

cot A= 1.5839.

sin A = .5338.

28

Example 2. Given a 50, b = 60, C90°, find A, B,

7. What is the height of a flagpole if at a horizontal distance of 200 feet from the foot of the pole the angle of elevation of its top is 19° 28' ?

8. A rope is stretched taut from the top of a building to the ground, and is found to make an angle of 58° 56' with the horizontal.

9. If a tree 74.3 feet high casts a shadow 42.6 feet long, how many degrees above the horizon is the sun?

10. A man walking on level ground finds, at a certain point, that the angle of elevation of the top of a tower is 30°. He walks directly toward the tower for a distance of 300 feet and then finds the angle of elevation of the top to be 60°. What is the height of the tower?

11. At a point, A, south of a tower the angle of elevation of the top of the tower is 60°. At another point 300 feet east of A the angle of elevation is 30°. What is the height of the tower?

12. The angles of a right triangle are 42° and 48°; the hypotenuse is 200 feet. What is the length of the perpendicular from the right angle to the hypotenuse?

13. The height of a gable roof is 20 feet, its width 42 feet. What is the pitch of the roof; that is, the angle it makes with the horizontal ?

14. From where I stand a tree 50 feet away has an angle of elevation of 43° 31'. From the same point another tree, 75 feet distant, has an angle of elevation of 32° 20'. Which tree is the taller and by how much?

29. Logarithms. The solution of right triangles as thus explained is simple in theory but may become laborious in practice because of the arithmetic computation involved. Fortunately we have in logarithms a device for simplifying such computation. The base of a system of logarithms is, in general, any arbitrarily chosen number.* In practice two systems are used: the Briggsian or common system of which the base is 10; and the Napierian system of which the base is e = 2.718.... The logarithm of a number to a given base (a) is the exponent of the power to which the base (a) must be raised to produce the number. Thus, if ax m, then x is the logarithm of m to the base a; written x = log。 m.

The word power is used here in its broader sense to include fractional and negative exponents. Defining fractional and negative exponents in such a way that the laws of exponents — aman — aman = am+n; (am)n -hold for negative numbers and fractions as well as for positive integers,

amn

* Some numbers, unity, for example, cannot be so used.

values of x may be found to satisfy, approximately at least, such an equation as a = b, no matter what values a and b may have. Thus, given any number, a, by raising it to a suitable power, p, and extracting a suitable root, q, of the result, we can obtain any other number, b; that is, Va=b. But this may be written a b or a b, where x =2, the

p

division of p by q being carried out to any desired number of decimal places. We then call a the logarithm of b to the base a.

30. The Common System. For purposes of computation the common system, base 10, is used. Let us form a table of powers of 10 and express the relations in terms of logarithms.

This table could be extended indefinitely in either direction. If we examine the table we notice that to produce a number between 1 and 10 we must raise the base 10 to a positive power between 0 and 1; to produce a number between 10 and 100, the exponent of the base must lie between 1 and 2; for a number between 100 and 1000, the exponent must lie between 2 and 3, and so on. In other words, the logarithm of a number between 1 and 10 lies between 0 and 1, and is, therefore, a fraction, always expressed as a decimal. The logarithm of a number between 10 and 100 lies between 1 and 2, or is 1 plus a decimal. The logarithm of

* Hereafter in this work we shall not write the base 10. Thus log 7 means log107. In general, however, except in works on trigonometry, if no base is written, e