In a similar manner the other ratios can be shown to be respectively unequal.

Ex. In this construction AP1 is taken equal to AP. Why does this not affect the generality of the proof?

(3) This property follows as a corollary from (1) and (2).

The trigonometric ratios for angles from 0° to 90° are arranged in tables. In some tables the calculations are given to four places of decimals, in others to five, six, or seven places. There are also tables of the logarithms of the ratios (or of the logarithms increased by 10),* which vary in the number of places of decimals to which the calculations are carried out. The student is advised to examine a table of the trigonometric ratios at this time. A good exercise will consist in finding the logarithms of some of the sines, tangents, etc., adding 10 to each logarithm, and comparing the result with that given in the table of Logarithmic sines, tangents, etc. [What are denoted as Natural sines and cosines in the tables, are merely the actual sines and cosines, which have been discussed above; the so-called Logarithmic sines and cosines are the logarithms of the Natural sines and cosines with 10 added.] A book of logarithms and trigonometric ratios is the principal help and tool in solving most of the problems in practical trigonometry; and hence, proficiency in using the tables is absolutely necessary. The larger part of the numerical answers in this book have been obtained with the aid of a five-place table. Those who use six-place or seven-place tables will reach more accurate results.

EXAMPLES.

1. Compare each of the ratios of RAL1 with the corresponding ratio of RAL.

2. Suppose that the line AR (Fig. 4) revolves about A in a counter-clockwise direction, starting from the position AM: show that, as the angle MAL

* These are usually called Logarithmic sines, tangents, etc.

increases, its sine, tangent, and secant increase, and its cosine, cotangent, and cosecant decrease. Test this conclusion by an inspection of a table of

Natural ratios.

3. Find by tables, sin 17° 40', sin 43° 25' 10", sin 76° 43', sin 83° 20′ 25′′, cos 18° 10', cos 37° 40′ 20′′, cos 61° 37', cos 72° 40' 30", tan 37° 40′ 20′′, tan 79° 37' 30", cot 42° 30', cot 72° 25' 30". Log sin 37° 20', Log sin 70° 21' 30", Log cos 30° 20' 20", Log cos 71° 25', Log tan 79° 30' 20", Log cot 48° 20' 40".

4. Find the angles corresponding to the following Natural and Logarithmic ratios:

14. Practical problems. The problems in this article are intended to help the learner to realize more clearly and strongly the meaning and the usefulness of the ratios which have been defined in Art. 12. The student is earnestly recommended to try to solve the first three problems below without help from the book. He will find this to be an advantage, whether he can solve the problems or not. If he can solve them, then he will have the pleasurable feeling that he is to some extent independent of the book; and he will thus be encouraged and strengthened for future work. Should he fail to solve them, he will have the advantage of a closer acquaintance with the difficulties in the problems, and so will observe more keenly how these difficulties are avoided or overcome. Throughout this course the student will find it to be of immense advantage if he will think and study over the subjectmatter indicated in the headings of the articles and make some kind of an attack on the problems before appealing to the book for help. If he follows this plan, his progress, in the long run, will be easier and more rapid, and his mental power more greatly improved than if he is content merely to follow after, or be led by, the teacher or author.

EXAMPLES.

1. Construct the acute angle whose cosine is . What are its other trigonometric ratios? Find the number of degrees in the angle.

The definition of the cosine of an angle shows that the required angle is equal to an angle in a certain right-angled triangle, namely, the triangle in which "the side adjacent to the angle is to the hypote

nuse in the ratio 2 : 3." Thus the lengths of this side and hypotenuse can be taken as 2 and 3, 6 and 9, 200 and 300, and so on. Taking the lengths 2, 3, (these numbers being simpler and, accordingly, more convenient than the others), construct a right-angled triangle AST which has side AS 2, and hypotenuse AT = 3. The angle A is the angle required, for cos A = }.

Now

=

ST=√32 – 22 = √5 = 2.2361.

Hence, the other ratios are

T

3

√5

A

2

S

FIG. 5.

ما

The measure of the angle can be found in either one of two ways, viz. : (a) by measuring the angle with the protractor; (b) by finding in the table the angle whose cosine is or .6667. The latter method shows that A 48° 11' 22". [Compare the result obtained by method (a) with the value given by method (b).]

2. A right-angled triangle has an angle whose cosine is, and the length of the hypotenuse is 50 ft. Find the angles and the lengths of the two sides.

50 Ft.

G

K

By method shown in Ex. 1, construct an angle A whose cosine is . On one boundary line of the angle take a length AG to represent 50 ft. Draw GK perpendicular to the other boundary line.

A

FIG. 6.

sin A

=

AG

The problem may also be solved graphically as follows. Measure angles A, G, with the protractor. Measure AK, KG directly in the figure.

24 Ft.

8 Ft.

3. A ladder 24 ft. long is leaning against the side of a building, and the foot of the ladder is distant 8 ft. from the building in a horizontal direction. What angle does the ladder make with the wall? How far is the end of the ladder from the ground?

Graphical method. Let AC represent the ladder, and BC the wall. Draw AC, AB, to scale, to represent 24 ft. and 8 ft. respectively. Measure angle ACB with the proMeasure BC directly in the figure.

tractor.

Method of computation.

BC=VAC2 - AB2 = √576 — 64 = √512 = 22.63 ft.

FIG. 7.

4. Find tan 40° by construction and measurement. With the protractor lay off an angle SAT equal to 40°. From any point P in AT draw PR perpendicularly to AS. Then measure AR, RP, and substitute

T

P

FIG. 8.

result thus obtained with the value given for tan 40° in

the tables.*

5. Construct the angle whose tangent is §. Find its other ratios. Measure the angle approximately, and compare the result with that given in the tables. Draw a number of right-angled, obtuse-angled, and acute-angled triangles, each of which has an angle equal to this angle.

6. Similarly for the angle whose sine is; and for the angle whose cotangent is 3.

7. Similarly for the angle whose secant is 23; and for the angle whose cosecant is 31.

8. Find by measurement of lines the approximate values of the trigonometric ratios of 30°, 40°, 45°, 50°, 55°, 60°, 70°; compare the results with the values given in the tables.

*The values of the ratios are calculated by an algebraic method, and can be found to any degree of accuracy that may be required.

If any of the following constructions asked for is impossible, explain why it is so.

9. Construct the acute angles in the following cases: (a) When the sines are 1, 2, 4, §; (b) when the cosines are,, 3, .3; (c) when the tangents are 3, 4, 3, 1; (d) when the cotangents are 4, 2, 3, .7; (e) when the secants are 2, 3, 1, 1, 4}; (ƒ) when the cosecants are 3, 2.5, .4,

10. Find the other trigonometric ratios of the angles in Ex. 9. Find the measures of these angles, (a) with the protractor, (b) by means of the tables. 11. What are the other trigonometric ratios of the angles: (1) whose sine is; (2) whose cosine is (3) whose tangent is (4) whose cotangent is; (5) whose secant is

; (6) whose cosecant is

?

b

12. A ladder 32 ft. long is leaning against a house, and reaches to a point 24 ft. from the ground. Find the angle between the ladder and the wall.

13. A man whose eye is 5 ft. 8 in. from the ground is on a level with, and 120 ft. distant from, the foot of a flag pole 45 ft. 8 in. high. What angle does the direction of his gaze, when he is looking at the top of the pole, make with a horizontal line from his eye to the pole?

14. Find the ratios of 45°, 60°, 30°, 0°, 90°, before reading the next article.

15. Trigonometric ratios of 45°, 60°, 30°, 0°, 90°. The ratios of certain angles which are often met will now be found.

By using the same figure it can be shown that