©<& UC.NRLF ^^aar a a^a k^ a^i^ y-?--^-^ m MEMOMAM Edward Bright ACADEMIC TRIGONOMETRY. PLANE AND SPHERICAL. BY T. M. BLAKSLEE, Ph.D. (Yale), Professor of Mathematics in the University of Des Moines. 3j*JC BOSTON: PUBLISHED BY GINN & COMPANY. 1888. J?9>,. Entered, according to Act of Congress, in the year 1»87, by T. M. BLAKSLEE, in the Office of the Librarian of Congress, at Washington. Typography by J. S. Cushino & Co., Boston. Pkbsswobk by GiNN & Co., Boston. PREFACE. r I ^HE purpose of this arrangement is to aid the memory by noting analogies. + and a^ have as spherical analogies x and cos a. Page 12 has page 13, and each "Law" has its analogy. In Spherical Trigonometry we note the determining groiqjs: side, +, co. function, and Z, — , function. It is hoped that the Introduction will fix the character- istics of Trigonometry. This should be accompanied by practical work, and occupy at least a week. T. M. BLAKSLEE. Des Moines, Ia. Note. It is convenient for examinations to have tables separate from formulas. The explanation of the use of tables should be with them. Two pages of model solutions and answers may be added. (Opin- ions asked on this point.) 797961 Digitized by tine Internet Archive in 2008 witii funding from IVIicrosoft Corporation littp://www.arcliive.org/details/academictrigonomOOblakricli INTKODUCTION. Def. Trigonometry is, etymologically, the Science of Meas- uring Triangles. Besides this, it now inchides the Science of Angular Functions. We first inquire, What is a function? then. What are the angular functions? A function of a variable is a second variable so related to the first that any change in the variable produces a change in the function. III. Oil in lamp and time it has burnt. Def. The functions of the angle between two straight lines are the six ratios of the sides of the right triangle formed b}" these lines and a perpendicular upon one of them from a point in the other. Notation, /i, hypotenuse ; o, opposite ; a, adjacent. h^ The ratios are, by definition, sine = sin = -• h .-. = /isin. sin cosine = cos = • h .-. a = /icos. cos tano;ent = tan = -• a ,'. = rttan, tan And their reciprocals. sin cos h a , - = -== tan. h a 8 ACADEMIC TRIGONOMETRY. By similar triangles, the functions are constants for a constant angle, but variables for a variable angle. The base line is the initial line ; the hypotenuse line, the terminus of the angle. Linear Representation. (1) If h=l, o = sin, a = cos. (2) If a= 1, o = tan. The transverse line is TT' through vertex and perpendicu- lar to initial line //'. sin = transverse projection of directed unit. (Unit 7i.) cos = initial projection of directed unit.* tan = transverse projection of h if initial projection be unity.f Since antecedent = consequent x ratio, also For sine and cosine, consequent = /i, ratio = function. Rule I. To obtain either side from /<, multiply by ratio, sine for o, cosine for a. Rule II. To obtain the sine from cosine, multiply by tangent. T'-\ h- T- II TA r t\ h I I a a > \^ h^ 1'.- — V" ;. ^ a a -h r "'ijT T T T Quadrants . IV and TT' divide the angular space about the vertex into four quadrants, numbered as in the figure. An angle is in the quadrant in which it. terminates. * The angle being the direction of its terminus, we may speak of the ratios as direction ratios. Since for the other acute angle of ratio triangle, sin = -, cos = -, and tan = - = cotangent = cot. h h .'. "co" means oj" complement. t If a circle be described with the unit base a as radius, o is a tangent. INTRODUCTION. The Terminal Values of the functions are as follows : I. II. m. IV. z 0°. 90^ 180^ 270°. sin + + - - sin + 1 -1 cos + - - + cos + 1 -1 tan + - + - tan GO GO The algebraic signs being determined thus : to right and up, + ; to left and down, — . PRACTICAL DEVELOPMENT. Wishing to calculate the distance IB to an object .B, start- ing from /, I laid off lA ± IB. At a distance AM=::: 1 from A I erected MN _L AI, deter- mining N by looking from A to B. I also measured AP, and drew PL ± AB. The last is not needed in measuring the distance ; in fact, AM might have been any distance, when IB could have been MNxAI unit L M found, as IB = AM The advantage of a table of tangents is, that we never have need to construct the small triangle. If lA = 1000 feet, and we have the tangent from a table, we have simply to move the decimal point three places, and we have IB at once. Two-Place Table. Take 10 inches as an hypotenuse, and, by aid of a protractor (or by consti'ucting an angle of 30°, geometrically, and then trisecting it by folding), construct the 10 ACADEMIC TRIGONOMETRY. values of sine and cosine f .-. tan = ^ ) for every 10°. Here \ cosy 10 inches = unit. .-. 0.1 inch = 0.01 unit. Evidently (arithmetically) function (180° - ^) = /(.4). The ratio triangles being equal, having h and A equal. EXAMPLES. z° 10 20 30 40 50 60 70 80 sin 17 34 50 64 77 87 94 98 cos 98 94 87 77 64 50 34 17 tan 18 30 58 84 1.19 1.73 2.75 5.67 1. Give functions, if o, a, A, are (1) 6, 8, 10; (2) 10, 24, 26; (3) 4, 7, 5, 8.5. 2. Solve the following : Z,o,aJi being (1) 20°, ?, ?, 100 ; (2) ?, 4, ?, 5; (3) 57', 4000, ?, ?; (4) 8.8", 4000, ?, ?. Note. If the greatest angular distance of Venus from the sun be 45°, what is its distance from that body as compared to that of the earth ? 3^ Canthe sines of 0°, 30°, 45°, 60°, and 90° be written, iVo, iVT, iV2, |V3. |V4? INTRODUCTION. 11 4. If A, B, and C be the angles, and a, 5, and c the opposite sides of a triangle, p the perpendicular from C to c, show that a sin jB = ^ = ?> sin ^. .- . a : b = sin A : sin B. (In words.) Do field work, using ratios to two places, sinl5 = i(0.17 + 0.34), sin 18 = sin 10 + 0.8 x 0.17 = 0.31. Note. If the greatest and least values of the maximum elongation of Mercury be 15° and 30°, what are its greatest and least distances from the sun ? Logarithmic Solutions. Though strictly Algebra, we give the logarithmic solutions thus far : log of sin = log — log 7i. . • . log o = log h + log of sin. log of cos = log a — log h. . • . log a = log h + log of cos. log of tan = log — log a. . • . log o = log a + log of tan. log sin = log of sin + 10, a:b = sin A : sin B, gives log sin A = log a -f colog b + log sin B. 12 ACADEMIC TRIGONOMETRY. PLANE. THE DEFINING EQUATIONS. (1) By T,, smA (2) By Tg, cos^ (3) By 7^2, tan^ a = h sin A, b = h cos yl, a = b tan ^4, h sin^ b _ cos^ a a b' ' tan^ (1) is the definition of the sine ratio. (2) is the definition of the cosine ratio. (3) is the definition of tlie tangent ratio. By T;, sin2 + cos2=l. .♦. a^ = Ji^ sin^, and b^ = Ir cos^. ... (4)i/i2 = a2-f-6'; (4)2 sin2-hcos2=l=cot^cotB. (4)i is the Pythagorean formnla. (5) By 7^4, sin ^ = cos 5, cos ^ = sin 5. (5) is the complementary formnla. Note. For construction of figure, p. 13, see p. 15. THE FUNDAMENTAL ANALOGIES. 13 C.B. SPHERICAL. tan. b THE FUNDAMENTAL ANALOGIES. / 1 \ T^ m • A Sin a . • T • 4 ' 1 sin a (1) By Ti, sin A = , sin a = sin h sin A, sin h = —- sin h sin ^ (2) By T3, cos^ = ^^, tan& ^ ^ *^ tan /a tan/i cos^, tan/i = (3) By T,, HnA tan a tan a = sin 6 tan A, sin 6 = tan?> cos^ tana tan J. sin 6 (1) is the sine analogy, or s'ui Ay. (2) is the cosine analogy, or cos Ay. (3) is the tangent analogy, or tan Ay. By T'4, VAiIIi, cos h = cos a cos 6, by (3) . (4) cos 7i — cos a cos 6 = cot A cot ^. -.iiJi The first is the Pythagorean analogy. Dividing sin^ (1) by cosJ5 (2), nsing (4). (5) sin A — cos B : cos 6, sin B = cos A : cos a. (5) is the complementary analog}'. Note. If a sphere of unit radius be described about Fas a centre, the faces will cut out a right spherical triangle having tlie sides a, h, and //. fi, 14 ACADEMIC TKIGONOMETKY. NEGATIVE ANGLES. Note. A point moving to the right generates a + distance, but moving back to the left tends to destroy this, and passing the origin generates a — distance. A straight line revolving in the order of the quadrants I, II, III, IV", generates a + angle, but revolving back tends to destroy this, and pass- ing the initial line generates a — angle. Rule. Changing the sign of an angle changes the sign of its sine, but not of its cosine. .'. changes that of its tangent. +z I. II. III. IV. -z -I. -II. -III. -IV. sin -f + - - sin - - + + cos + - - + cos + - - + tan + - + - tan - + - + Since the terminus is changed across the initial line II', but not across the transverse line TT'. Thatis, /'=IV, 7/'= III, III'=U, IV'=J. FUNCTIONS OF nr±A. Rule. If an acute angle be added to or subtracted from an even number of quadrants, the functions of the resulting angle are equal arithmetically in value to the like-named functions of the acute angle ; but if an acute angle be added to or subtracted from an odd number of quadrants, the func- tions of the resulting angle are arithmetically equal in value to the co-named functions of the acute angle. By the equality of the eight possible ratio triangles, and the fact that for an even number of quadrants a and o are CONSTRUCTION. 15 the same as for A, but for an odd number they are inter- changed. Note. By revolving 1^ and 1^^ through any number of right angles, one rotation changes sine to cosine, two restores, and so on. CONSTRUCTION. (1) Lay off from the vertex Fof a right trihedral a unit on each edge (F// being edge of rt. Z). (2) Through the extremity of one of the acute edges, as ^i, pass a plane ± to the other acute edge VA^ thus : Draw BH± VH, then HA ± VA, lastly join AB. (By Geom.) BAH is the plane measure of dihedral having edge VA. (3) Through the otlier extremities H^ and ^3, pass planes II \joA,BJI,, .-. ± to VA. (4) By p. 12, the parts of the nine right triangles are as given. Note. Napier's Circular Parts are : Tlie two sides about the right angle, the complements of the opposite angles, and the complement of the hypotenuse. His rules are : Rule I. The sine of the middle part is equal to tlie product of the tangents of the ar^jacent parts. Rule II. The sine of the middle part is equal to the product of the cosines of the opposite parts. 16 ACADEMIC TRIGONOMETP.Y. By I sin a = sin A sin h = tan b cot B (1)1 sin 6 = sin ^ sin A: cos^= sin B cosa tan a cot A tan b cot A (3) (5) ""'' "^- ""• ^ ""'^ ^ - ^"" " """ " (2) I cos J5= sin ^ cos 6 = tan a cot A | (4) I cos A = cos a cos 6 = cot ^ cot i? 1(4) I. By (Comp, Ay.) An oblique angle and its opposite side are in the same quadrant. II. By (P. Ay.) h < 90° when a and b are in the same quadrant. h > 90° when a and b are in different quadrants. EXAMPLES INVOLVING RIGHT TRIANGLES. Solve in order of formulas. Each triangle furnishes nine examples. 5,C,? s,1,t 1,c,t s,c, t ?,?, ? h a b A B 1. 10.40 5.43 8.87 31° 30' 58° 30' s = sin Def . 2. 37.57 27.81 25.26 47° 45' 42° 15' c = cos Def. 3. 7855. 6967.8 3627.4 62° 30' 27° 30' t = tan Def. 4. 50 30 40 36^ 52' 10" 53° 7' 30" P = Pyth. Theo. 5. 13 5 12 22° 37' 10" 67° 22' 50" Comp.=co-relation. 6. The distance of the moon being 7i, and earth's radius a A = 57' 2" ; find h. For the sun, A = 8.8". 7. What is the length of the horizontal shadow of the Washington Monument, when the altitude of the sun is 50^? 8. What is the radius of the circle of latitude on which you live ? 9. (a) The angle of elevation of the top of a spire 500 feet distant is measured and found to be 13°. What is its height [above the instrument] ? (h) The elevation of base of spire is 9°. What is its height ? EXAMPLES. 17 10. The elevation of the top of a spire is 45°, and at a point 100 feet farther away, 36° 52'. What is the height? y t-V Spherical Right Triangles. First find a from A and /i, then A from a and li. That is, solve in the order that the formu- las are given (p. 13). a b h A B 1. 20° 4' 22" 59° 33' 43" 23° 27' 29" S.Ay. 2. 12° 56' 43" 34° 14' 45" 23° 27' 31" 3. 57° 21' 33" 59° 33' 43" 23° 27' 29" C.Ay. 4. 43° 20' 58" 44° 9' 38" 13° 34' 30" 5. 45° 7' 49" 85° 45' 2" 45° 12' 33" T. Ay. 6. 38° 1'40" 139° 24' 50° 14' 8" 7. 22° 7' 15" 69° 30' 13" 71° 4' 20" P. Ay. 8. 2° 59' 22° 21' 51" 22° 33' 12" 9. 43° 35' 46° 59' 58° 50' Comp. Ay. 10. 22° 15' 7" 27° 28' 38" 73° 27' 11" The right ascension i2, declination cZ, and longitude Z, of the sun form a right triangle of which these are 6, a, and h ; A being the obliquity of the ecliptic. 11. i = 214° 14' 45"; find i2 and d. 12. 72= 18 hrs. 44 min. 50 sec. ; find L and d. 13. i2 = 4hrs. 38 min. 0.88 sec, d = 22° 7' 13.7"; find L and A. REMARKS AND QUESTIONS. Many points rest directly upon page 12. Thus pages 13, 14, 18, 19, and 23 in great part. As the last of 24, and most of 25, require 19 and 20, the given order has been followed. It is very important for the student to observe as to what rests directly on pages 12 and 13. 18 ACADEMIC TRIGONOMETRY. Give the values of the functions of: 60°, 120°, 150°, 225°, -30°, -60°, -1210°, 350°, 440°, 1000°, 1234°. How are the functions of 115° related to those of 205°? Reduce {x^ + y^)cos 720°- 2xy sin 540°. When is it sufficient to consider angles in (1) I, (2) I and II, (3)1, -IV, (4)V, -VI, ? VALUES OF ONE FUNCTION IN TERMS OF ANOTHER. By P. Theo., V = sin^ + cos^ = 1. (1) .-. sin= Vl — cos^, cos = Vl — sin''^. Note. Prove this without P. Theo., and thus prove the theorem. By P. Theo., f-^\ = rec. cos^ = tan^ + 1. \cosy 1 . , tan . • . cos = — =zi=:^ , sin = tan cos = — -• Vl + tan^ a/1 + tan^ Give the value of each function in terras of the others. 1 . sin 30° = ^ ; find the other functions. 2. cos 45° = I V2 ; find the other functions. 3. tan ^1= 1, 2, 3, ... ; find otlier functions. Hint. If tan = 3, rec. cos^ = 10. .-. cos = , sin = — ^• VTO VlO 4. Given one function of 90° ; find the others. 5. (1) 2 sin = cos (2) tan = 2 sin. (3) sin = COS. (4) sin = tan. Hint on (2). — = 2 sin. .'. cos = I. Z= 60°. cos Hint ON (1). sin = r, 2x= Vl — x-. .-. a: = sin = ^V5. 6. cosnan2 + sin2cot2=l. 7. In rt. sph. A, sin^/i = sin^o + sin^a [cos^o] (2d P. Ay.). = sin-a H- sin^o [cos^a]. Its limit, /r = a" + o^ FUNCTIONS OF SUM. 19 (a) FUNCTIONS OF SUM. ■-^^A COS. A COS. B.—sin.A sin..B. Directly from figure and page 12, sin (^ + -B) = sin ^ cos JB + cos ^ sin ^. (1) cos (^ 4- B) = cos J. cos 5 — sin ^ sin ^. (2) Dividing sine by cosine, then both terms of fractions by cos A cos B, tan(J[ + iB)=*^^^^i±i^^. l-tanu4tanJ5 (6) Double Angle. IfA = B. sin 2 J. = 2 sin A cos A. cos 2A = cos^A — s'm^A. 2tan^ tan 2 A l-tan^^ (c) Sum and Difference of Sines. If A-\-B A-B=D. (3) (1) (2) (3) S, and A = ^ + ^, B = ^-^. 2^2 2 2 sin ^ = sin — cos -- -f- cos — sin — sin 5 = sin f cos :5 _ cos - sin - 2 2 2 2 sin^ + sinJB=2sin-cos- 2 2 (1) sin ^ — sin -B = 2 cos— sin — 9 9 (2) 20 ACADEMIC TRIGONOMETRY. (a) FUNCTIONS OF DIFFERENCE. sin (A-B) = BinlA + ( -5)] = sin^4cos(— 5) + cos^siii(— i^). By rule for negative angles, page 14, sin (A — B) = sin ^ cos JB — cos^ sin jB. (1) cos(^ — -B) = cos^cos JB-f sin^sin^. (2) tan(^-^)= ^^^^-;^^-^ . (3) ^ ^ 1 + tan^tanjB ^ (6) Half Angle. By double angle formulas, COS"* — h sin''— = 1 . 2 2 cos^ sin^— = cos^. 2 2 .*. having the sum and the difference of sin^ and cos^. . A 1 — cos^l x^x A cos — 2 =aI^^^- (2) ''CO + sm tan^ = Ji^^^. (3) 2 \l + cos^ ^ ^ (c) Sum and Difference of Cosines. , S D . S . D cos A = cos — cos sin - sin — 2 2 2 2 „ S D , . S . D cos JB = cos — cos f- sin— sin — 2 2 2 2 cos^+cos5= 2 cos— cos— * (1) S' T) cos-4 — cosJB = — 2 sin— sin — (2) EXERCISES. 21 EXERCISES. 1. Find sine and cosine of 90° q: J, 180°:f ^, •-.. 2. Find siu3^= sin(2 J. + ^) = 3 sin^- 4 siuM, cos 3 J. = cos {2A-\-A) = 4: cos^A — 3 cos A. 3-7. If sin 30° = |, (3) find sine and cosine of 15° ; (4) of 45° ; (5) of 22^° ; (6) of 67^° ; (7) as answer, find those of 90°, or (67^°+ 22^°). 8. sina;4- cosflj= Vl + sin2ic. 9. If Z tan t be read "the angle whose tangent is ^," show that Ztan J +Ztani = 45° ; also, ^ + ^= 90° if ^ = Z sinf and 5 = Z sin |. 10. Find the area of a regular dodecagon inscribed in a circle of radius 12. ADDITIONAL. 1. If a + 6 = ^-t-^/i, findsinA 2. Find tan— by bisecting A. Hint. If the bisector divide sin^ into two parts, x and >j, x:y = 1 : cos A. :. tan - = ■' = ^+y = sin^ 2 cos A 1 + cos A 1 + cos A 3. If tan- tan- tan- = 1, find sin a and cos a in terms of A '^ Jj the sine and cosine of h and c. 3'. If tan^tan(^45°-:^^cot^=l, find the sine and cosine of each angle v, P, and E^ in terms of those of the other two. (Due to Prof. O. Stone.) 3". tan (^45° _ ^'^ = Vl - sin P : Vl + sin P* 22 ACADEMIC TRIGONOMETRY. Va / ^''\ / \ &r^ I I . _i I — J I sin. A sin. B COS. B COS. A 4. Find sum and difference of sines and cosines directly from figure and page 12. Hint. The diagonals of a rhombus bisect at right angles. Half sum = greater — half difference. sin ^ + sin 5 = 2 YY^ =: 2 sin ^ VY'. 5. (cos^ - cos 5)2 + (sin ^ - sin jB)2= 4 sin^^— -?. 6. cos2^ = (l-tanM) : (l + tan^^). 7. As27rr = 360%r° = — =57.3. TT How many degrees in L5r? Ans. 270^ What is the length of 80°, r being the unit? Ans. 57.3 8. If 1^= cos^+ -sin^, .-. cos J. = ^ "^^ ~^ , and sin ^ 1 2 -, 1^ X Is being 1(4+5), and the binomial formula z holding for such quantity, find sin(J. + i5), cos(yL+J5), sin ^ + sin 5, etc., including cos 2^, sin 3^, cos 4yl, -f. = +V — 1. 1a= initial cos A + transverse sin A. 9. The earth's radius subtends an angle of 57' at the moon ; what is the distance (by 7) ? The moon's apparent diameter is 31' ; what is it in miles (by 7) ? EXERCISES. 23 10. Construct the figure of page 13 : (1) When one side is in I, and the other in II. (2) When both are in II, .-. h in I. 11. Construct the figure of page 19 : (1) When A and B are in I, but {A + B) in II. (2) When A is in I, and B in II. (3) When both are in II. 12. Find tan {A+B) from the figure of page 19. Hint. The base of figure is now to be taken as 1. Note. The "12 additionals " are only for the leaders of the class, and not for the body. Part of 8, though sometimes found in Algebra, seems more nearly in place here. \a = cos J. + • sin Ay (1) I5 ^ cos J5 + • shi B ; (2) multiplying, 1(^+5) = (cos A cos J5 — sin J. sin B) +. (sin A cos B -f cos A sin B), but l(A+5) = cos (^ + J5) + • sin {A + B). .-. the usual formulas. If instead of multiplying (1) and (2) we add them, observing (p. 22) that a journey of 2 cos — in direction - causes the same change of posi- tion as the two journeys 1^ and 1^, we have the usual formulas for sin ^ + sin J5 and cos A + cos B. ^W^" 24 ACADEMIC TllIGONOMETRY. PLANE. Law of Sines. In any plane triangle, the sides are propor- tional to the sines of the opposite angles. . a : 6 = sin ^ : sin B. By definition of sine, a sin JB = J) = 6 sin A. Law of Cosines. The square of any side of a (pi.) triangle is equal to the sum of the squares of the other two sides, minus twice their product by the cosine of the included angle. Pythagorean Proposition : a^ = p^ -\-m^, b^=p^-\-n^. .*. a^ — h^ = iw — n^. (In words.) a^-b'' = {c-7iy-ri' = c''-2cn. Definition of Cosine : nz=b cos A. - ,'. substituting and transposing term 5^, a^=b'--\-c'-2bccosA. Law of Tangents. The sum of any two sides of a (pi.) tri- angle is to their difference as the tangent of one-half the sum of the opposite angles is to the tangent of one-half their difference. ' By law of sines and theory of proportion, ft + & _ sin ^ + si" I^ ^ tan |^(^1 -f- B) a — b sin^ — sin5 t^ni{A— B) LAWS. 25 SPHERICAL. Law of Sines. In any spherical triangle, the sines of the sides are proportional to the sines of the opposite angles. ^ 71 X) m D^--n^ By sin Ay., sin asmB= sinp = sin b sin A. .'. sin a : sin b = sin A : sin B. Law of Cosines. The cosine of any side of a (sph.) triangle is equal to the product of the cosines of the other two sides, plus the product of their sines by the cosine of their included angle. Pythagorean Analogy : cos a = Qosp cos m, cos b — cosp cos n. .'. cos a : cos6 = cosm : cos?i. (In words.) cosa : cos6 = cos(c— ??): eos?i = cosc-f- sine tan 7i. Cosine Analogy : tann = tan 6 cos^. Substituting and transposing /ac^or cosb, cos « = cos & cose + sin 6 sine cos ^. % For Analogy of Law of Tangents, see the limit of Napier's (2), page 29. 26 ACADEMIC TRIGONOMETHY. FUNCTIONS OF HALF ANGLES. Plane. By law of cosines, cos A = ^^ — - — ^ = —^ -26c 26c Problem. To substitute this value in licosJ^, and thus find the functions of the half angles in terms of the sides. If a4-6 + c=2s, .'. a + 6 — c= 2(s — c), a— 6 + c=2(s — 6), 6 + c — a= 2(s — a). 1— cos^ = ' = ^^ -^ 26c 26c ^ 2(^-6)2(g-c) 26c Similarly, 1 + cos ^ = 2s{s-a) ^ be sm A l(s-b)(s-c) - .A ll-cos^ — =A -^^ — -] from sm— =\ . 2 \ 6c 2 \ 2 A Is(s-a) . .4 |l + — = \ -^^ ^: from cos - = -\ — - 2 \ 6c ' 2 \ cos^ COS tan^^JI^-^)(--^). 2 \ s{s-a) That is, the sine of half an angle of a plane triangle is equal to the square root of half the sum of the three sides minus one of the including sides, into the half sum minus the other including side, divided by the product of the including sides. Give for cosine and tangent. Ex. Find the value of tan— directly from page 12, and Wentworth's Geometry, page 250. HALF ANGLES. 27 If r be the radius of inscribed circle, area = rs. By Geometry, an -ea = Vs(s — a) (s — h) (s — c) , ^,^ \ (s-a){s-h){s-c) \ s By figure, tan— = 2 s — a FORMULAS FOR HALF ANGLES. Spherical. By law of cosines, . cos a — cos h cos c cos A = • sin b sin c As in Plane, sin b sin c + cos 6 cos c — cos a 1 — cos A = sin b sin c cosa — cos(6 — c) / V* i? \ = V 1 (clif . of cos) — sin b sin c _ 2 sin(g — 6) sin (8 — c) sin 6 sine Similarly, sin 6 sine .-. the analogy here is the same as in law of sines, A_ j8in(s — sm__ , ^ -fe)sin(s-c). sin b sin c ^ _ Isin s ^^„— , sin(s — a) cos — — ' ^^ ^' sin b sin c tan-= | sin(.s-6)sin(s-c) 2 "V sinssin(s — a) That is, the cosine of one-half of either angle of a spheri- cal triangle is equal to the square root of the sine of one-half of the sum of the three sides, into the sine of one-half this 28 ACABEIkUC TRIGONOMETRY. sum minus the side opposite the angle, divided by the product of the sines of the including sides. Giv'e sine and tangent in words. ^ LAW OF COSINES FOR ANGLES. FUNCTIONS OF HALF SIDES. If A'B'C be the polar of ABC, by first law of cosines and Geometry, cos^ = — cos jB cos (7+ sin^sinCcosa. In words, cos A 4- cos B cos G cosa = / . , sin i5 sm G H — GosS cos{S — A) sin sin B sin C cos 2=\ - B) cosjS - C) sin B sin C In words. Note Analogies. THE GAUSS EQUATIONS. Bin i(A-\-B) cOs^c = cos^{a — b) cos^C. cos^{A + B) Gos^c = cos|(a + b) sin^C. sin i{A — B) sin^c = sin^(a — b) gos^O. cosi{A — B) s'm^c = sin ^(a + 5) sin ^ C. fH^±B) flc= /i(a±6) no. Rule I. sin in (1) gives — in (3) and conversely. cos in (1) gives -f in (3) and conversely. Rule II. Functions have same names in (2) and (3). Functions have co-names in (4) and (1). Note. The rules also hold for I, II, III, and IV. napier's pkoportions. 29 To prove (1), smi{A -h B)= smf^ + ^\ Now sin ( — H — ) = sin— cos — h cos — sin — \2 2j 2 2 2 2 _ / siu(s — 6) sin(s — c) / sing sin(8 — &) \ sin & sin c \ sin a sin c / sin g sin (s — a) / sin(s — a) sin (.9 — c) \ sin b sin c \ sin a sin c sin(s — 6) (7 , sin(s — a) C = — ^^ ^cos 1 ^^ ^cos- sin c 2 sine 2 sin(s — 6)-fsin(s — a) C .^ ^ - \ = — \ L^ ^ ^cos— (bv sum of sines) sine 2 ^ - . 2 sinf s ^^!^— icos C cos — ; sine 2 u 4. a -\-h c but s -^^- = — 2 2 2 sin — cos — C cos .*. Q. E. D. £i • C C 2 . 2 Sin— COS— ^ 2 2 By the Gauss Equations and (Div. Ax.) we have NAPIER'S PROPORTIONS. (1) sini(a + b):sini(a - b) : icotiC : tani(A- B). (2) cosi(a + b):cosi(a - b) : : cot^C : tani^A -\- B) . (3) sin^(^-f-5):sin^(yl-5)::tanic : tan^Ca - b). (4) cosi(^ + J5):cosi(^-S)::tan^c :tan^(a + b). Theorem 1 . The sine of one-half the sum of either tivo sides of any spherical triangle is to the sine of one-half their differ- ence as the cotangent of one-half the angle ichich they include is to the tangent of one-half the difference of the angles opposite. 30 ACADEMIC TKIGONOMETRY. Theorem 2. The cosine of one-half the sum of either two sides of any spherical triangle is to the cosine of one-half their difference as the cotangent of 07ie-half the angle ivhich they include is to the tangeyit of one-half the sum of the angles opposite. Theorem 3. The sine of one-half the sum of either two angles of any spherical triangle is to the sine of one-half their differ- ence as the tangent of one-half the side which they include is to the tangent of one-half the difference of the sides opposite. Theorem 4. The cosine of one-half the sum of either two angles of any spherical triangle is to the cosine of one-half their difference as the tangent of one-half the side ichich they include is to the tangent of one-half the sum of the sides opposite. Note the analogies: sine giving — , cosine, + ; side giv- ing "co"/, angle giving function. EXAMPLES UNDER OBLIQUE TRIANGLES. Given (any three parts, one being a side) : I. One side and two angles, L. of Ss. II. Two sides and Z opposite one of them, L. of Ss. III. Two sides and the included Z, L. of Ts. or L. of Cs. IV. Three sides. Formulas for half A or L. of Cs. V. Sph. Three A. Formulas for half sides. I. {a,A,B) (b,A,B) (c,A,B) (a,A,C) (b,A,C) (c,A,C) {a,B,C) (h,B,C) (c,B,C). II. (a,b,A) {a,b,B) {a,c,A) {a,c,C) (b,c,B) {b,c,C). III. (a, 6,0) {a,c,B) {b,c,A). a b c A B C 3. 1686 960 2400 33° 34' 39" 18° 21' 21" 128° 4' 4. 40 34 45 83° 53' 15" 57° 41' 25" 38° 25' 20" 5. 10 12 14 44° 24' 65" 57° 7' 18" 78° 27' 47" 1. 6 8 10 36° 52' 53° 8' 90° 2. 2 Vo 45^ 60° 75° ASTRONOMICAL TRIANGLES. 31 6. Find the distance between two objects (supposed inac- cessible) by calculating the distance to each by right triangles, also by law of sines, then measuring the angle between these distances. Measure the distance to test your answer. 7. If in (4) the unit be one mile, and CA be east and west line, what is the direction of each vertex from the others ? 8. An object when viewed from the ends of an east and west line, of length 34, bears N. 51° 35' W. and N. 6° 7' E. What is its distance from each, and from the straight line joining them ? a b c A B C 1. 2. 3. 4. 70O 4' 17" 20O16' 147° 5' 33" 69° 34' 56" 63° 21' 26" 56° 18' 165° 5' 18" 58^16' 22" 66° 18' 33° 1'36" 70° 20' 20" 81° 38' 20" 20° 10' 110° 10' 50° 10' 10" 70° 9' 38" 550 54' 133° 18' 63° 31' 20" 114° 18' 30" 70° 20' 40" 50^30' 8" As the chief applications are to Astronomy, we give THE FIRST ASTRONOMICAL TRIANGLE PZH. P is north pole of equator. Z is zenith (pole of horizon) . IT is the heavenly body. Altitude of pole = AT' = latitude of N\ place. The distances of a body from any great circle and its pole are complementary. PZ = co-latitude. HP— polar distance = co-declination. HZ= zenith distance = co-altitude. Z. Z= azimuth. AP— hour or time angle. Example. Calculate the following for a place, latitude 45°, and for the longest day of the year. 32 ACADEMIC TKIGONOMETllY. (a) The azimuth of sun at setting and rising. (b) Time of rising and setting. (c) The greatest altitude of the sun. (d) Time when a vertical object casts a horizontal shadow of its own length. THE SECOND ASTRONOMICAL TRIANGLE. Before considering this triangle involving two great circles and their poles, we will consider a point referred to any great circle and its pole. Let F (any point of this great circle) be the origin, and consider the hemisphere limited by the great circle of F. P b P h E E f^ ^ \ ^ /^A^ \ M \ ' In V ^ ^ \/ 1 yj.'AQ \x /W^ ■^ 1 V v^ V Let P be the pole, and EQ, the great circle of reference, H the heavenly body ; VH is, by Geometry, _L to great circle of F cutting it in (r ; denote PQ by 5, HG by a. VF^:^ dii-ect co-ordinate D, FQ = co. D. (Fig. 2.) FH= transverse co-ordinate T, HP =co. T. From the rt. A HPG and HVF (C. Ay.), tan h — cot T sin Z). .*. tan r = cot & sin Z). tana (Tan. Ay.), tan(co. i>) = cotD (P. Ay.) , cos DcosT= cos VH. sin 6 (1) (2) (3) (4) In the second triangle PP'H, we have two systems of co- ordinates : Right ascension {R) and declination (d), when REDUCTION OP OBSERVATION. 33 referred to the equator EQ^ but latitude I and longitude L^ when referred to the ecliptic EG. Since, if the former be D and T, the latter are Z)' and T', if F, the common point of the two circles, be the origin : tan h = cot d sin R. sin 6 _ tan R sin 6' • (1) (2) by (3) above, taniv tan 6' = cot Z sin iv. (3) cos I cos L = cos d cos R. (4) PF= E, the obliquity of the ecliptic ; P' = co.i ; P=90+i2. The reduction of an observation from the equator to the ecliptic is of so much importance that one example is worked through and back. M 81° 48' 42.4" L 85° 45' 2.00" d 68° 27' 54.5" I 45° 7' 48.98" sinU 9.9955499 " sinL 9.9988044 coid 9.5961411 cot I 9.9980251 tan b^ 9.5916910 tan 6 9.9968295 h' 21° 20' 1.58" b 44° 47' 27.11" e 23° 27' 25.53" e 23° 27' 25.53" b=b'-\-e 44° 47' 27.11" b'=b-e 21° 20' 1.58" sin b 9.8478940 sin b^ 9.5608631 1 : sin b^ 0.4391369 1 : sin 6 0.1521060 tanU 0.8419622 tan L 1.1289931 tanL 1.1289931 tanjR 0.8419622 L 85° 45' 2.00" JB 81° 48' 42.4" sin L 9.9988044 sin JB 9.9955499 cot 6 0.0031705 cot b' 0.4083090 tan I 0.0019749 iand 0.4038589 I 45° V 48.98" d 68° 27' 54.5" cos I 9.8484953 cose? 9.5647190 COS L 9.8698113 cos It 9.1535877 COS / COS L 8.7183066 COS dcosR 8.7183067 34 ACADEMIC TRIGONOMETRY. QUADRANTAL TRIANGLES. By Geometry, the polar of a right triangle is a quadrantal triangle. If A'B'C be a rt. A, and ABC its polar, then, by Napier's'Bules, sin^= tan ^ cot & = sin a sin^. sin B — tan A cot a = sin 6 sin H. cos a = — tan B cot H= cos A sin h. cos 6 = — tan^ cot jEr= cos J5 sin a. cos H— — cot a cot 5 = — cos A cos ^. That is, Napier's Rules would hold for a quadrantal tri- angle, if the circular parts be the two angles about the quadrant, the complements of the opposite sides, and the complement of the hypotenuse angle, giving THREE SECONDARY ANALOGIES. sin^ tan-B . tan^ sm a = •> cos a = ? tan a — — - — • smlT tan^ sin ^ Rule. Those formulas that contain a co-function of H have a negative sign. I. The same as for right triangles. Proof by I, of rt. A, if ^^90% a'^OO^, ^'^90°, .-. a^90°. ' II'. The reverse of II, for right triangles. Proof. If A and B in same quadrant, .-.a' and 6' in same. By II, /i'< 90% .-. H> 90°. q. e. d. Proof of Rule. There are but two hypotheses : (a) A and B in the same quadrant ; (6) A and B in different quadrants. If (a), by II", H> 90°, .-. co-function H -. But, by I', the other two factors are alike in signs. ,-. sign of formula must be —. QUADRANTAL TKIANGLES. 3t5 If (6), by II, H< 90°, .-. co-function of // -|-. But by I, the other two factors are alike in sign. .-. sign of formula must be — . .*. q. e. d. EXAMPLES UNDER QUADRANTAL TRIANGLES. 1. In what latitude does the sun rise in the northeast at Glimmer solstice ? 2. Find time of sunrise and sunset at your school on the longest day of the year. 3. Rt. A. Form a table of times when the shadow of the side of the east school door-way will coincide with an east and west crack in the hall floor. Closing Note. Establish the definitions of sine and cosine without similar triangles, thence laws of sine and cosine and Pythagorean proposition. Then give trigonometric proofs of proportions concerning oblique lines, sides of a triangle opposite equal angles, and converse. Greater side opposite greater angle, and converse. 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British Mail: All engaged in the practical work of education will appreciate tliese Manuals, as they are calculated to save the master much precious time and labor, and to give his students the benefit of progressive and carefully thought- out exercises. Wentworth & Hill's Exercises in Geometry. 12mo. Cloth. 255 pages. Mailing Price, 80 cents ; Introduction Price, 70 cents. nPHE exercises consist of a great number of easy problems for beginners, and enough harder ones for more advanced pupils. The problems of each section are carefully graded, and some of the more difficult sections can be omitted without destroying the unity of the work. The book can be used in connection with any text- book on Geometry as soon as the geometrical processes of reason- ing are well understood. select propositions from it to supple- ment every stage of our work. Amelia W. Platter, Hiyh School, Indianapolis, Ind. : I find the sub- ject so carefully graded, that I can Analytic Geometry. By G. A. Wentworth. Revised edition. 12mo. Half morocco, xii + 301 pages. Mailing Price, $1.35; for introduction, $1.25; allowance in exchange, 30 cents. rpiiE aim of this work is to present the elementary parts of the subject in the best form for class-room use. The connection between a locus and its equation is made per- fectly clear in the opening chapter. MATHEMATICS. 71 The exercises are well graded and desired to secure the best mental training. By adding a supplement to each chapter provision is made for a shorter or more extended course, as the time given to the subject will permit. The book is divided into chapters as follows : — PART I. Plane Geometry. I. Loci and their Equations; II. The Straight Line; IIL The Circle; IV. Different Systems of Co-ordinates ; V. The Parabola ; VI. The Ellipse ; VII. The Hyper- bola; VIIL Loci of the Second Order; IX. Higher Plane Curves. PART II. Solid Geometry. I. The Point; II. The Plane; III. The Straight Line ; IV. Surfaces of Revolution. Dascom Greene, Prof, of Mathe- matics and Astronomy, Rensselaer Polytechnic Institute, Troy, N.Y. : It appears to be admirably adapted to tbe use of beginners, and is espe- cially rich in examples for practical application of the principles of each chapter. The full and clear explana- tion of first principles given in the opening chapter is a new and highly commendable feature of the work. {Nov. 11, 1886.) Geo. D. Olds, Prof, of Mathematics, University of Rochester, N. Y. : It is a most admirable little book. The author falls into line with what I believe to be the best modern ten- dency in text-books, — the avoidance of bulk and complexity. (JVo?J. 11, 1886.) J. L. Patterson, Teacher of Mathe- matics, Laivrenceville School, 2i.J. : I do not know of any text-book for beginners in this subject which can compare with it for class-room use. E. A. Paul, Prin. of High School, Washington, D.C. : I think it is to be commended for the same clearness of statement and simplicity of ar- rangement for which the author's other works are noted, and believe it to be especially adapted for ad- vanced pupils, in high schools and academies, who wish to know some- thing of the mysteries of loci and conic sections, and who have only a limited time for the work. {Nov. 12, 1886.) Jos. J. Hardy, Prof, of Mathe- matics, Lafayette College, Easton, Pa. : The professor's experience has taught him what are the points which the boys find obscure, and he has generally been successful in devising a good way of elucidating these ix)ints. . . . Teachers will find it a very helpful manual. {Jan. 4, 1887.) E. Miller, Prof, of Mathematics, University of Kansas, Laivrence : As a book for beginners, it is admi- rable in all its arrangements and features. The great number of problems scattered through it will largely relieve it of that abstract analysis which is so often a terror to students. The book is, like the other works of Professor Wentworth, a good thing. (Nov. 18, 1886.) 72 MATHEMATICS. A Treatise on Plane Surueying. By Daniel Cabhart, C.E., Professor of Civil Engineering in the West- ern University of Pennsylvania, Allegheny. Illustrated. 8vo. Half leather, xvii + 498 pages. Mailing Price, $2.00; for introduction, $1.80. rpmS work covers the whole ground of Plane Surveying. It illustrates and describes the instruments employed, their ad- justments and uses; it exemplifies the best methods of solving the ordinary problems occurring in practice, and furnishes solutions for many special cases which not infrequently present themselves. It is the result of twenty years' experience in the field and technical schools, and the aim has been to make it extremely practical, having in mind always that to become a reliable surveyor the student needs frequently to manipulate the various surveying instruments in the field, to solve many examples in the class-room, and ,to exercise good judgment in all these operations. Not only, therefore, are the different methods of surveying treated, and directions for using the instruments given, but these are supplemented by various field exercises to be performed, by numerous examples to be wrought, and by many queries to be answered. Chapter I. Chain Surveying. " II. Compass and Transit Surveying. " III. Declination of the Needle. " IV. Laying Out and Dividing Land. " V. Plane Table Surveying. " VI. Government Surveying. " VII. City Surveying. Including the Principles of Levelling. " VIII. Mine Survejring. Including the Elements of Topography. The following Tables have been added: Logarithms of num- bers ; Approximate equation of time ; Logarithms of trigonometric functions ; For determining with greater accuracy than the pre- ceding ; Lengths of degrees of latitude and longitude ; Miscellaneous formulas, and equivalents of metres, chains, and feet; Traverse; Natural sines and cosines ; Natural tangents and cotangents. The judicial functions of surveyors, as given by Chief Justice Cooley, are set forth in an appendix. The work is published just as this Catalogue goes to press, so that full notices cannot be given. Send for the special circular. MATHEMATICS. 77 Byerly^s Syllabi. By W. E. Byerly, Professor of Mathematics in Harvard University. Each, 8 or 12 pages, 10 cents. Syllabus of a Course in Plane Trigonometry. Syllabus of a Course in Plane Analytical Geometry. Syllabus of a Course in Plane Analytic Geometry. (Advanced Course.) Syllabus of a Course in Analytical Geometry of Three Dimensions. Syllabus of a Course on Modern Methods in Analytic Geometry. Syllabus of a Course in the Theory of Equations. Elements of the Differential and Integral Calculus. With Examples and Applications. By J. M. Taylor, Professor of Mathematics in Madison University. 8vo. Cloth. 249 pages. Mailing Price, $1.95; Introduction Price, $1.80. rpHE aim of this treatise is to present simply and concisely the fundamental problems of the Calculus, their solution, and more common applications. Its axiomatic datum is that the change of a variable, when not uniform, may be conceived as becoming uniform at any value of the variable. It employs the conception of rates, which aifords finite differen- tials, and also the simplest and most natural view of the problem of the Differential Calculus. This problem of Jinding the relative rates of change of related variables is afterwards reduced to that of finding the limit of the ratio of their simultaneous increments ; and, in a final chapter, the latter problem is solved by the principles of infinitesimals. Many theorems are proved both by the method of rates and that of limits, and thus each is made to throw light upon the other. The chapter on differentiation is followed by one on direct integra- tion and its more important applications. Throughout the work there are numerous practical problems in Geometry and Mechanics, which serve to exhibit the power and use of the science, and to excite and keep alive the interest of the student. 78 MATHEMATICS. The Nation, New York: It has two marked characteristics. In the first place, it is evidently a most carefully written book. There is nothing vague or slipshod in it. Nearly every sentence, certainly every theorem, seems to have heen constructed with a strenuous effort to give it clearness and precision. This constant attention to the form of expression has enabled the author to be concise without becoming ob- scure. We are acquainted with no text-book of the calculus which com- presses so much matter into so few pages, and at the same time leaves the impression that all that is neces- sary has been said. In the second place, the number of carefully se- lected examples, both of those worked out in full in illustration of the text, and of those left for the student to work out for himself, is extraordi- nary. From this point of view, those teachers and pupils who are accus- tomed to or prefer a different text- book, would still do well to provide themselves with this, regarding it merely as a collection of examples and without any reference to the text. Elementary Co-ordinate Geometry. By W. B. Smith, Professor of Physics, Missouri State University. 12mo. Cloth. 312 pages. Mailing Price, $2.15; for introduction, $2.00. VITHILE in the study of Analytic Geometry either gain of *' knowledge or culture of mind may be sought, the latter object alone can justify placing it in a college curriculum. Yet the subject may be so pursued as to be of no great educational value. Mere calculation, or the solution of problems by algebraic processes, is a very inferior discipline of reason. Even geometry is not the best discipline. In all thinking, the real difficulty lies in forming clear notions of things. In doing this all the higher faculties are brought into play. It is this formation of concepts, therefore, that is the essential part of mental training. And it is in line with this idea that the present treatise has been composed. Professors of mathematics speak of it as the most exhaustive work on the sub- ject yet issued in America ; and in colleges where an easier text- book is required for the regular course, this will be found of great value for post-graduate study. Wm. G. Peck, Prof, of Mathe- matics and Astronomy, Columbia College : I have read Dr. Smith's Co- ordinate Geometry from beginning to end with unflagging interest. Its well compacted pages contain an im- mense amount of matter, most ad- mirably arranged. It is an excellent book, and the author is entitled to the thanks of every lover of mathe- matical science for this valuable con- tribution to its literature. I shall recommend its adoption as a text* book in our graduate course. MATHEMATICS. 79 Academic Trigonometry : piane and spherical. By T. M, Blakslbb, Ph.D. (Yale), Professor of Mathematics in the University of Des Moines. 12mo. Paper. 33 pages. Mailing Price, 20 cents; for introduction, 15 cents. rpHE Plane and Spherical portions are arranged on opposite pages. The memory is aided by analogies, and it is believed that the entire subject can be mastered in less time than is usually given to Plane Trigonometry alone, as the work contains but 29 pages of text- The Plane portion is compact, and complete in itself. Examples of Differential Equations. By George A. Osborne, Professor of Mathematics in the Massachu- setts Institute of Technology, Boston. 12mo. Cloth, vii + 50 pages. Mailing Price, GO cents; for introduction, 50 cents. A SP^RIES of nearly three hundred examples with answers, sys- ■'^ tematically arranged and grouped under the different cases, and accompanied by concise rules for the solution of each case. Selden J. Cofl&n, lately Prof, of I Its appearance is most timely, and it Mathematics, Lafayette College : I supplies a manifest want. Determinants. The Theory of Determinants: an Elementary Treatise. By Paul H. Hanus, B.S., recently Professor of Mathematics in the University of Colorado, now Principal )f West High School, Denver, Col. 8vo. Cloth, viii + 217 pages. Mailing Price, $1.90; for introduction, $1.80. rpHIS book is written especially for those who have had no pre- vious knowledge of the subject, and is therefore adapted to self-instruction as well as to the needs of the class-room. The subject is at first presented in a very simple manner. As the reader advances, less and less attention is given to details. Throughout the entire work it is the constant aim to arouse and enliven the reader's interest, by first showing how the various concepts have arisen naturally, and by giving such applications as can be presented without exceeding the limits of the treatise. The w^ork is sufficiently comprehensive to enable the student who has mastered the volume to use the determinant notation with ease, and to pursue his further reading in the modern higher algebra with pleasure and profit. GENERAL LIBRARY UNIVERSITY OF CALIFORNIA— BERKELEY RETURN TO DESK FROM WHICH BORROWED This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. l8Apr'55KV ,955 Ut 220ct:56WJ REC'D Lb OCT 8 1956 FEB U 1^*3-' 21-100m-l,'54(1887sl6)476 YB 56843 797961 UNIVERSITY OF CALIFORNIA LIBRARY