12. Assuming that the sun passes directly overhead, trace the change in the length of the shadow of an object from dawn to sunset. Which trigonometric function do you think of in this problem? 13. Assuming the results of Exs. 10 and 11, derive from them the variation of the tangent from 0° to 360° and its values at each of the angles mentioned in Ex. 11. Do the same for ctn a, sec α, csc α. 51. Reading of Tables. Sine and Cosine of - 0 and 90° +0. In order to find the sine (or any other trigonometric function) of an angle we consult the tables. In the tables the values of the sine, for example, are printed only up to 45°. To find the sine of an acute angle greater than 45° we make use of the relation sin α = cos (90° — α). The tables are arranged to facilitate this by having the angles above 45° printed at the bottom of the page, and the column headings changed from sine to cosine, etc. (See Tables, p. 22.) If we wish to find the sine of an angle greater than 90°, we must find a way to express the sine in terms of some function of an acute angle less than 45°. In Chapter III this has been done for obtuse angles. The purpose of what follows is to make a similar reduction for any angle, positive or negative. The following four relations are true for every angle 0 (positive, negative, zero): 90+0 B FIG. 54 P(a,b) (b) When 0° < 0 < 90°, place the angles 0, 0, and 90° +0 on the same coördinate axes, with the origin as center and a convenient radius r > 0, and de(a,-b) scribe a circle cutting the terminal sides of 0, 0, 90° +0, in P, Q, R. Let the coördinates of P be (a, b); then those of Q are (a, — b), and those of R are (—b, a), since the right triangles OAP, OAQ, RBO are congruent. Then, by § 49, sin 0 = b/r, cos 0 a/r; sin (0) b/r, cos ( 0) = a/r; and sin (90° + 0) = a/r, = — = = cos (90° +6)=-b/r. Therefore sin (0) = sin 0, cos (0) 0) = cos 0, sin (90° + 0) = cos 0, cos (90° + 0) == - sin 0. (c) When 90°, 180°, 270°, verify by direct substitution. (d) When 90° < 0 < 180°, or 180° < 0 < 270°, or 270° < 0 <360°, the formulas are proved by a figure drawn as in (b). The formulas (1)-(4) are thus proved for all values of between 0° and 360° (0° included). angle ẞ be (See § 41.) but since Finally, if a is any angle whatever, there is one tween 0 and 360° (0° included) such that ẞa. We have just seen that formulas (1)–(4) hold for ß; aß, we have also − a ≈ — ß, 90° + α = 90° + ß. (See § 41.) It follows that (1)-(4) hold for a, since any trigonometric function has the same value for any two congruent angles. 52. Reading of Tables. Sine and Cosine of 180° ±0 and 270° ± 0. The relations tabulated below are true for every angle 0. Proof: The four relations in black-face type have been proved above; to prove the others, we proceed as follows: (1) Let α =- - 0, then 90° - 0 = 90° + α, and sin (90° — 0) = sin (90° + α) = cos α = cos (— 0) = cos 0. = == - sin α = sin (0) = sin 0. 180° 0, then = whence (2) Cos (90°-0) = cos (90° +α) = (3) Let α = 90° Ө · 90° + α, sin (180° — 0) = sin (90° + α) = cos α = cos (90° — 0) : = sin α = - sin (90° — 0) and cos (180° - 0) : = cos (90° + α) cos 0. = sin 0; (4) Let α = 90° +0 and make use of the formulas for 90° + a to obtain formulas for 180° + 0. (5) Let a = 180° - 0 and make use of the formulas for 90° + a to obtain formulas for 270° — 0. (6) Let α = · 180° + and make use of the formulas for 90° + a to obtain formulas for 270° +0. F 53. Extension to Other Functions. The relations tabulated below are true for every angle 0, except as noted: Proof: The relations in black-face type have been proved above; to make the others depend on these, use the relation tan α= sina cosa and the fact that the last three functions are the reciprocals of the first three in reverse order. The whole body of relations in this table, which have now been proved for all angles 0, except as noted below, ‡ may be remembered by the two following "rules of thumb." Ꮎ 1. Determine the sign by the quadrant in which the angle would lie if 0 were acute; the result holds whether is acute or not. 2. In case of 90° +0 or 270° + change the name of the function to the cofunction; in case of — 0 or 180° ± 0 do not change the name of the function. Example 1. sin(175°) = sin(180° — 5°)=+ (rule 1)sin(rule 2)5°. cos 175° = Example 3. cos (90° + 85°) =—(rule 1)sin(rule 2)85° == tan 300° = tan (270° + 30°)=— (rule 1)ctn (rule 2)30° = — √3. Example 4. tan 300° = tan (180° +120°) =+(rule 1) tan (rule 2) 120° *When @ = an odd multiple of ± 90° it has no tangent or secant. † When /3. = an even multiple of 90° it has no cotangent or cosecant. The tabulated relations are all true if is not a multiple of 90°; they fail only in the cases mentioned in the preceding footnotes. See §§ 8, 49. EXERCISES XXII.-READING OF TABLES-REDUCTION TO 1. Express the following as functions of acute angles not greater than 45°. Make use of congruent angles whenever advantageous : 3. Make a tabular form of 5 columns and 16 rows, and at the head of the columns, beginning with the 2d, enter the words sine, cosine, tangent, cotangent. In the first column, beginning with the 2d line, enter the following angles: 172° 26', - 153° 18′, 253° 12', 208° 25', 285° 32', — 312° 18′, 389° 15', — 416° 27', 462° 50', 502° 11', 552° 37', - 618° 42', - 700° 24′, 1000° 10′. Use a table of trigonometric functions. 650° 14', functions of acute angles as in Ex. 2: (a) log(-cos 161° 11'). (c) log(sin 217° 17'). (c) tan 264° 46'. (d) ctn 128° 14'. (g) sin 142° 25'. (h) cos 275° 23'. (b) log sin 161° 11'. (d) log(cos 252° 48′). [Note that the numbers in parentheses in (a), (c), and (d) are positive; if the minus sign were absent, each of them would be negative. Negative numbers have no real logarithms.] 6. Compute the values of the following expressions by logarithms: (a) 2.35 sin 148° 23'. (b) 24.8 cos 160° 40'. (c) 16.2 cos 320° 45'. 7. Find the components on the axes of a force of magnitude 5.74 (lb.) which makes an angle of 215° 20' with the positive end of the x-axis. 8. A force is indicated by stating its magnitude (in pounds) and its direction (i.e. the angle it makes with the positive end of the x-axis). Find the components on the axes of the forces indicated below: (a) (4.17, 128°). [This means magnitude 4.17, angle 128°.] (b) (24.8, 250° 10′). (d) (51.4, 141° 25'). (c) (5.72, 310° 35'). 9. Find the magnitude and the direction of a force whose components on two perpendicular axes are F2 = 25.46, Fy = 38.72. 10. Find the magnitude and the direction of a force whose components are F 12.8, F1 = 6.45. PART II. GRAPHS OF TRIGONOMETRIC FUNCTIONS 54. Plotting of Values. A table of a few values, such as those of Exs. 10 and 11, p. 63, furnishes enough data to construct a fair graph of the functions sine and cosine. The table on p. 15 also may be used; it contains more than is really needed except for exceedingly accurate drawing. For example, if x denotes the angle and y denotes its sine, we have to plot the curve represented by the equation y sin x. = On a sheet of cross section paper draw a pair of rectangular axes Ox and Oy. Before plotting any points it is necessary to decide how many spaces shall represent a unit length and how many degrees shall be represented by the unit length, in order to get all the desired points on the paper. In Fig. 55, one unit on the horizontal scale is chosen to represent 60°, which is convenient to the size of the paper; but any other unit might as well have been chosen. We then plot the points corresponding to the angles indicated, and draw through them a smooth curve, keeping in mind the general behavior of the sine as given in the first of the tables of Ex. 10, p. 56. In plotting curves it is of advantage in many ways to make the horizontal and vertical scale units the same, and this should be done if not too inconvenient.* * If we were to take the two scale units the same in plotting the curve y = sin where the unit angle is the degree, one arch of the curve would be 180 units long and only one unit high. |