FUNCTIONS OF ACUTE ANGLES 25 23. Changes in the Functions. If we suppose ZAOP, or x, to increase gradually to 90°, the sine MP increases to M'P', M"P", and so on to OB. That is, the sine increases from 0 for the angle 0°, to 1 for the angle 90°. Hence O and 1 are called the limiting values of the sine. Similarly, AT and OT gradually increase in length, while OM, BS, and OS B gradually decrease. That is, As an acute angle increases to 90°, its sine, tangent, and secant also increase, while its cosine, cotangent, and cosecant decrease. T" S S P T x M'M'M A If we suppose x to decrease to 0°, OP coincides with OA and is parallel to BS. Therefore MP and AT vanish, OM becomes equal to OA, while BS and OS are each infinitely long and are represented in value by the symbol . Similarly, we may consider the changes as x increases from 0° to 90°. Hence, as the angle x increases from 0° to 90°, we see that sin x increases from 0 to 1, cos x decreases from 1 to 0, tan x increases from 0 to ∞, cot x decreases from ∞ to 0, sec x increases from 1 to ∞, csc x decreases from ∞ to 1. We also see that sines and cosines are never greater than 1; secants and cosecants are never less than 1; tangents and cotangents may have any values from 0 to ∞. In particular, for the angle 0°, we have the following values: For the angle 90° we have the following values : By reference to the figure and the table it is apparent that the functions of 45° are never equal to half of the corresponding functions of 90°. Thus, sin 45° = 0.7071, cos 45° = 0.7071, tan 45° = 1, sec 45° = 1.4142, Exercise 13. Functions as Lines 1. Draw a figure to show that sin 90° = 1. Draw a figure to show this. 2. What is the value of cos 90°? Draw a figure to show this. 3. What is the value of sec 0°? 4. What is the value of tan 90°? 5. What is the value of cot 90°? 6. As the angle increases, which increases the more rapidly, the sine or the tangent? Show this by reference to the figure. 7. If you double an angle, does this double the sine? Show this by reference to the figure. 8. If you bisect an angle, does this bisect the tangent? Prove it. 9. State the angle for which these relations are true: sin x cos x, tan x = cot x, Show this by reference to the figure. secx CSC x. 10. If you know that sin 40° 15'=0.6461, and cos 40° 15'=0.7632, and that the difference between each of these and the sine and cosine of 40° 15' 30" is 0.0001, what is sin 40° 15' 30"? cos 40° 15' 30"? 11. If you know that tan 20° 12' is 0.3679, and that the difference between this and tan 20° 12' 15" is 0.0001, what is tan 20° 12' 15"? 12. If you know that cot 20° 12' is 2.7179, and that the difference between this and cot 20° 12' 15" is 0.0006, what is cot 20° 12′ 15′′? 13. If you know that tan 66.5° is 2.2998, and that the difference between this and tan 66.6° is 0.0111, what is tan 66.6°? 14. If you know that cos 57.4° is 0.5388, and that the difference between this and cos 57.5° is 0.0015, what is cos 57.5° ? Draw the angle x for which the functions have the following values and state (page 11) to the nearest degree the value of the angle: 1.66. +1.5 = 0. sin x 33. Find the value of sin x in the equation sin x х Which root is admissible? Why is the other root impossible? 24. cot x = 4.0. 29. secx = 1.7. CHAPTER II USE OF THE TABLE OF NATURAL FUNCTIONS 24. Sexagesimal and Decimal Fractions. The ancients, not having developed the idea of the decimal fraction and not having any convenient notation for even the common fraction, used a system based upon sixtieths. Thus they had units, sixtieths, thirty-six hundredths, and so on, and they used this system in all kinds of theoretical work requiring extensive fractions. 15 60 For example, instead of 17 they would use 1 28', meaning 128; and instead of 1.51 they would use 1 30′ 36′′, meaning 130+ The symbols for degrees, minutes, and seconds are modern. 36 3600 We to-day apply these sexagesimal (scale of sixty) fractions only to the measure of time, angles, and arcs. Thus and 3 hr. 10 min. 15 sec. means (3 + 10 + 3150) hr., 15 60 3° 10' 15" means (3+10 +350)°. 3600 In medieval times the sexagesimal system was carried farther than this. For 10 20 30 45 example, 3 10′ 20′′ 30′′ 45iv was used for 3+ + + + Some 60 602 608 604 writers used sexagesimal fractions in which the denominators extended to 6012 Since about the year 1600 we have had decimal fractions with which to work, and these have gradually replaced sexagesimal fractions in most cases. At present there is a strong tendency towards using decimal instead of sexagesimal fractions in angle measure. On this account it is necessary to be familiar with tables which give the functions of angles not only to degrees and minutes, but also to degrees and hundredths, with provision for finding the functions also to seconds and to thousandths of a degree. Hence the tables which will be considered and the problems which will be proposed will involve both sexagesimal and decimal fractions, but with particular attention to the former because they are the ones still commonly used. The rise of the metric system in the nineteenth century gave an impetus to the movement to abandon the sexagesimal system. At the time the metric system was established in France, trigonometric tables were prepared on the decimal plan. It is only within recent years, however, that tables of this kind have begun to come into use. 25. Sexagesimal Table. The following is a portion of a page from the Wentworth-Smith Trigonometric Tables: The functions of 41° and any number of minutes are found by reading down, under the abbreviations sin, cos, tan, cot. Decimal points are usually omitted in the tables when it is obvious where they should be placed. The secant and cosecant are seldom given in tables, being reciprocals of the cosine and sine. We shall presently see that we rarely need them. Since sin 41° 2' is the same as cos 48° 58' (§ 8), we may use the same table for 48° and any number of minutes by reading up, above the abbreviations cos, sin, cot, tan. Trigonometric tables are generally arranged with the degrees from 0° to 44° at the top, the minutes being at the left; and with the degrees from 45° to 89° at the bottom, the minutes being at the right. Therefore, in looking for functions of an angle from 0° to 44° 59', look at the top of the page for the degrees and in the left column for the minutes, reading the number below the proper abbreviation. For functions of an angle from 45° to 90° (89° 60'), look at the bóttom of the page for the degrees and in the right-hand column for the minutes, reading the number above the proper abbreviation. Exercise 14. Use of the Sexagesimal Table From the table on page 28 find the values of the following: In the right triangle ACB, in which C = 90°: 16. Given c = 27 and A = 41° 3', find a. 17. Given c = 48 and A = 42° 4', find a. 31. A hoisting crane has an arm 30 ft. long. When the arm makes an angle of 41° 3' with x, what is the length of y? what is the length of x? 32. In Ex. 31 suppose the arm is raised until it makes an angle of 41° 5' with x, what are then the lengths of y and x? 30 41°3′ 20 33. From a point 128 ft. from a building the angle of elevation of the top is observed, by aid of an instrument 5 ft. above the ground, to be 42° 4'. What is the height of the building? 34. From the top of a building 62 ft. 6 in. high, including the instrument, the angle of depression of the foot of an electric-light pole is observed to be 41° 3'. How far is the pole from the building? |