EXERCISES 1. Telescopes at the end of a base line, 250 feet long, on the deck of a ship are turned upon a distant fort. The lines of sight of the telescopes are found to make angles of 89° 12′ and 89° 40′ with the base line. Find the distance from the ship to the fort. Ans. 2.39 miles. Suggestion. In Fig. 77, let B and C be the positions of the telescopes and A the position of the fort. 2. The diameter of the moon subtends an angle of 31'5" at the earth. The moon is approximately 240,000 miles from the earth. Find the diameter of the moon in miles. Ans. 2170 miles. REVIEW EXERCISES In Exercises 1 to 13 transform the first member into the second. 7. (sin a cos B cos a sin ẞ)2 + (cos a cosẞ + sin a sin ẞ)2 = 1. 8. sin2 a cos2ß + sin2 a sin2ß + cos2 a 9. tan10+ sec1 0 = 2 sec2 0 tan20 +1. = 1. sin40 cos2 (1 + sin 0) 18. If x2 + y2+ z2 = r2, x = r cosa, y = r cos B, and z = r cos y, show that cos2 a + cos2 8+cos2y= 1. 19. For what values of k will tank - sin2 k = tan2 k • sin2 k? 4 CHAPTER V FUNCTIONS OF SUMS AND DIFFERENCES OF ANGLES 65. In the previous chapters, we have worked with, and established the relations between, the functions of a single angle. But, in solving oblique triangles and in many of the applications of trigonometry to other subjects, formulas are used which are derived from the functions of the sums or differences of angles. These functions are expressed in terms of the functions of the angles and are as follows for the sine and cosine: Formulas [13] and [14] are often called addition formulas, and [15] and [16] subtraction formulas. 66. Derivation of the formu las for the sine and cosine of the sum of two angles. Let LAOB = α and L BOC Fig. 78, each of which is acute, and so chosen that a+B= ZAOC is less than 90°. In order that the functions of a, B, and a + B may be involved in the same formula, we may form right triangles which have a, B, and a + ẞ as acute angles. Draw PH10A, Choose any point P in the terminal side OC. PD10B, DK LOA, and DL1PH. ▲ KOD is similar to ▲ LPD, since their sides are perpendicular each to each. Then LPD = a. HP KD+LP KD LP + Now multiply numerator and denominator of KD OP by OD, the common side of the two triangles of which KD and OP are sides LP respectively. Also multiply in the same way by PD, the common side of triangles DOP and LPD. Then 67. Derivation of the formulas for the sine and cosine of the difference of two angles. - Let ZAOB = a and ZCOB = B be the two acute angles, Fig. 79. Then angle AOC = a - B. For reasons similar to those given in the preceding article, choose any point P in the terminal side OC of (a 8). Draw PH LOA, PRIOB, RD1OA, and PELDR. A DOR is similar to ▲ PER and ERP = α. [16] OH = = OP OD + EP OD EP OP = OD OR = + EP RP OR OPRP OP' cos (a - B) = cos a cos ẞ + sin a sin ẞ. In the proof of [15] and [16], it was assumed that a > B. Now which is the same result as was obtained before. Exercise. Show that cos (a - ẞ) gives the same result whether a > ẞ or a < ß. 68. Proof of the addition formulas for other values of the angles. - In Art. 66 formulas [13] and [14] were proved when a, B, and a + B are each less than 90°. They are, however, true for all values of the angles. (1) Suppose that a and B are acute and such that a = Y, where and y are each less than 45°. sin [(90° — $) + (90° − y)]=sin [180° − (p+y)] sin (+7) = sin cos y + cos o sin y. Substituting for the functions of and y their values in terms of the functions of a and ẞ, That is, the formula for sin (a + B) is true when (a + B) is an angle in the second quadrant and a and ẞ as stated. In the same way we may show that the formula for cos (a + B) is true for values of the angles as given above. (2) Suppose that a is in the second quadrant and B in the third, such that a = 90° + and B = 180°y. On this assumption, = cos 0, cos a = cos (90° + ¢) - sin &, sin a = sin (90° + $) sin ẞ=sin (180° + y) = sin (a+8)=sin [(90° + ø) + (180° + y)] = sin [270° + (+y)] γ. = = cos (+ y) - cos cos y + sin & sin Substituting for the functions of of the functions of a and B, and y their values in terms sin (a + B) = - (sin a) (-cos B) + (-cos a) (-sin ẞ) = sin a cos ẞ + cos a sin ß. In the same manner it may be shown that the addition formulas are true for any angles. 69. Proof of the subtraction formulas for all values of the angles. Since the addition formulas are true for all values of a and B, they are true when -ẞ is put for B. Then sin (a) sin [a + (-8)] = sin a cos (-8) + cos a sin (-8), and = B) = cos [a + (-8)] = cos a cos (-ẞ) — sin a sin (−B). cos (a That is, the subtraction formulas are true in general. EXERCISES Prove that formulas [13] and [14] are true in the following cases: 1. a in the fourth quadrant and ẞ in the first. 2. a in the third quadrant and ẞ in the third. 3. a in the first quadrant and ẞ in the third. Solve the following exercises by means of the addition and subtraction formulas, assuming a and 6 less than 90°. = 4. Find the sine and cosine of 90° by assuming that 90° 60° + 30°. Solution. sin 90° = sin (60° +30°) = sin 60° cos 30° + cos 60° sin 30°. Substituting the values of the functions of 30° and 60°, sin 90° = √3. √3 + } · } = { + } = 1. cos 90° |