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M.A omuif's nxivBurrr KINGSTON R. UGLOW & COMPANY ■^. -J •* :ili r ■■' •;* •\ ■ A BRIEF TREATISE ON PLANE TRIGONOMETRY MOSTLY ON THE PRACTICAL SIDE AND INTENDED FOR PRACTICAL SCIENCE STUDENTS BY N F. DUPUIS, M.A., qcnif'i inivnaiTT K^ NGSTON R. UGLOW Sc COMPANY 1902 ok Sit -^^3 7 •H ■Bwr«l •ooordluif U> Act of the Pwrllament of Canada. In th. year ont thouMd nine hundred and two. by B. Ueu>w ft Co.. at the DepMt- ment of Agriculture. 7 mt irt- 48875 ! hh^mh PREFACE. This little book has been written as a help to students in practical trigonometry in the School of Mining. If it should prove to be helpful to anyone else, so much the better. It is not constructed along usual lines. The exercises are many and varied, and are largely practical, while some of them are proofs of minor and readily obtained theorems. Exercises in transformations which may be beautiful and interesting, but are not of practical use, are not many. The student is encouraged to work with natural functions, as in the experience and opinion of the writer these are more direct, more manageable with small angles, and fully as expedi- tious as logarithmic methods, and in some cases more so. From a perusal of some modem works on the subject, a student would rise with the idea that logarithms are an essential and necessary part of trigonometry, and that nothing can be done without them. He who forms such an idea has faUed to grasp the nature of the subject, and to understand the force and meaning of the trigonometric functions. In practical life men should learn to do their work with a minimum of appliances, and a small table of natural functions is all that is required in practical trigonometry. For these reasons logarithms and logarithmic methods are relegated to the latter portions of the work, and the theory of logarithms is supposed to be learned where it should be learaed, in connection with arithmetic and algebra. \- V I I i I ; i /^ i r .. \ CONTENTS. Decimal Approximation Errors Measures and Units of Measure Radian Measure Connection between Urits Trigonometric Ratios . Inter-relation of the Functions Complement and Supplement The Triangle .... Cosine Formula Ambiguous Case . Area of Triangle Orthogonal Projection . Applications to Forces and Velocities Addition Theorems Changing Sums to Products, etc. Logarithmic Formulas . Transformation of Cosine Form Inverse or Circular Functions Trigonometric Constructions . Miscellaneous Exercises . Answers to Exercises . / 7 9 9 lU 12 14 17 19 21 23 23 28 29 32 36 37 38 42 43 46 51 \\\ ! i : i I I I ! I ii ,o il I Ml ill! "^^r A it; \ ^'ili A BRIEF TREATISE ON PLANE TRIGONOMETRY. I. Of Decimal Approximation.— In the practical meas- urement of lengths, angles, weights, etc., results are not usually integral or definitely fractional, but expressible by a series of decimal figures. Thus, in measuring the distance between two points with a scale divided to tenths of an inch, we might ep^^^iroate the hundredths, and find the distance to be 3.14 inches say. If the measure were graduated to hundredths of an inch, we might estimate to thousandths approximately, and find the distance to be 3.141 inches. But in any case the final decimal figure in our result is only an approximation, and the expression IS only an approximate value for the distance. It is obvious that the more decimal places we include, other things being the same, the closer is the approximation. Example.— The ratio of the circumference of a circle to its diameter, or of the seiT>i-circumference to the radius is 3.1415926.... (1) to seven decimal places. 3.14 is the approximate value to two decimals, and 3.1416 to four decunals; because, as we reject all after the 5, and 59 is nearer to 60 than to 50, we change the 5 into 6. The majority of quantities occurring in trigonometry are of this nature, that is, they are decimal approximations carried to four, five, six, etc., decimal places. 3. Errors.— All practical measurements are affected by two 8 TRSATISB ON PLANE TRIGONOMETRY. H t I ill ■\.i sources of errors : (1) Errors of oonstruotion in the instmment employed, and (2) errors in making and recording the observa- tion. And when the error of a decimal approximation is \esf than the errors due to observation, the approximation is sufli ciently close for practical purposes. Thus, it is not possible without the aid of a microscope tc measure a distance to within a thousandth of an inch ; hence, such a distance expressed in inches to the nearest unit in th( third decimal place is sutficiently accurate as far as expression goes. NoTB. — At this work will be largely practical, and will deal continuallj with four, five and six place decimals, students should learn and practise contracted methods of working with decimals, and study to become expert in their use. EXERCISE I. 1. Find to six decimal places the length of the circumference of a circle whose radius is 1 mile or 5280 feet. 2. In regard to Ex. 1, find the error in feet resulting fron taking 3.1 instead of 3.1415926. Also find the errors fronr taking 3.141 ; 3.1416. 3. In employing a six decimal approximation for 3.141592( what number would you take, and why 1 4. In expressing a length in miles, how many decimals an required to give it to the nearest foot 1 To the nearest inch 1 5. In expressing an area in acres, how many decimals an required to give it to the nearest square foot 7 To the nearest square inch t 6. The area of a field is given as 18.7415 acres, which is trulj expressed to the fourth decimal figure. What is the greatest possible error in the expression 1 Give result in s {uare feet. 7. Multiply 1.4142136 by 0.7071068 to seven decimals in the product, using contracted multiplication. 8. Multiply 0.0471404 by 21.213204 to five decimals. 9. The sides of a rectangle are 13.3412 and 24.467 ft. Fine the area to the nearest square inch. i l;i TR11TI8I ON PLANE TRIOONOMSTRV. 9 Find to three 12. 13. 10. The jeasure 0.7312 is taken 0.085 times, deoiraals the length measured. 11. Divide 1 by 2.30258509 to seven decimals (The result of this divwion i. the modulu, of our common system of logarithms.) /••«"• Divide 180 by 3.1415926 to six decimals. m.e quotient give- the number of degree, in one radian.) Work out to three decimals the value of (7960 X 5280 x 3. i 4 1 5926) + 1 296000. (The result i. the number of feet in one i«cond of arc of Utitude on the earth s surface. ) 3. Measures and Units of Measure.-Every measure must be expressed in units of its own kind. Thus, lengths are expressed m units of length, such as mile, foot, inch. etc. ; time m uniU of time, as year, day. hour, etc. So also angles must be expressed in angular units Angle i^ generated by the rotation of a variable line about a fixed pomt. As the rotation may be with the hands of a clock or against them, angle may be negative or positive. Usually but not necessarily, rotation against that of the hands of a clock IS taken as positive. A line which, starting from any given direction, makes a complete rotation, returning to its original di .ction, measures the simplest unit angle, the circuraangle. This is subdivided thus : 1 circumangle = 2 straight angles = 4 right angles- 360^ Then 1 right angle = 90° • 1° = 60' , 1 ' » 60". Degrees, minutes, seconds of angle are marked ° ' " Th's division of angle is very ancient, and is known as the sexayeaimal or ,legree measure. It forms the basis of the major- ity of trigonometric tables. 4. Radian Measure; natural measure; circular measure ot an angle. It is proved in geometry that in the same circle the lengths of arcs are proportional to the angles which they subtend at the 10 TREATISE ON PLANE TRIGONOMETRY. ! II M i centre. So that if « be the length of an arc, and be the angle which it subtends at the centre, « = mO, where m is an arbitrary constant. If m is taken to be the radius, then a^rO.... (2) and the resulting value of is taken to be the radian measure oi the angle subtended by s. Hence the length of an arc is the angle which it auhtends in reuiians at the centre x the reditu. If » = r, then ^=1. Therefore the unit of radian measure, or one radian, is tht angle at the centre subtended by an arc equal in length to tht radius. 5. Connection between the Units.— If c denotes the length of the semi-circumference of a circle, we know that c=rx 3.1415926. .. Or, denoting 3.1415926 by ;r, as is usual, c= 7tr. Hence 7t is the radian measure of two right angles. 180°= n^, denoting radian by ^. Hence and r=jg^= OrOl 74.53, 180 1^* = = 57°.29578.. n (3; .ill I These numbers, 0.017453, the multiplier by which to change degrees into radians, and 57.29578 which changes radians t< degrees, should be carefully remembered. Thus, 64"= 64x0.01745 = 1.1 1680. ..radians and 0^.71654 = 0.71654 x 57.29578 = 4r.045. . . . For small angles, say less than 1°, the length of the chore may be taken for that of the arc in practical work without anj material error, and the error reduces rapidly as the angle diminishes. )he angle irbitrary (2) easure of >tends in a, is the h to the lotes the at (3) ) change kdians to he chord hout any be angle !' I II ,*-^ 'i im m \ .11: TREATISE OK PLANE TRIOONOMETRY. 11 EXERCISE II. 1. Express 36° 14' 20" in degrees and decimals of a degree. 2. Express 25' 15". 34 as a decimal of a degre*. 3. Express 64". 35 as a decimal of a degree. 4. Express 3°.8472 in degrees, minutes and seconds. 5. With radius 1 mile, find the length in inches of the arc which subtends an angle of 1". 6. The earth's radius being 3,980 miles, find the number of seconds in 1 foot of are on its surface. 7. Express 1^0472 in degrees. 8. Express 50° 47' 57" in radians. 9 A house at the distance of a mile subtends a horizontal angle of 35' 44". Find the dimensions of the house. 10. A tree is known to be 76 feet high, what angle will it subtend at the distance of a mile ? 11. How far from the eye must a disc 2 feet in diameter be held that it may just hide the sun, the angular diameter of the sun being 32'? 12. How far must a man 6 feet tall go away from camp that he may subtend an angle of 50' ? 13. A man 5 feet 8 inches tall standing upon the opposite bank of a river subtends an angle of 18'. Wl:ai is the breadth of the river 1 14. A wheel 12 feet radius revolves 12 times per minute. Find the rate per second at which the rim travels. 15. A and £ are on the same meridian. A's latitude is 32° 14' 12" N., and B's is 27° 15' 40" N. Find the distance from A to JB, the earth's radius being 3,980 miles. 16. If the difference in latitude of A and B (Ex. 15) be 1° 6' 49", and their distance apart be 77.3 miles, find the earth's diameter. 17. The earth's distance from the sun is 9'j,000,000 miles, ^and it makes its annual revolution in 365.2422 days. Find its velocity in miles per second. 11! SH lii' m \ ill il 12 TREATISE ON PLANE TRIGONOMETRY. 18. The moon's distance from the earth is 237,000 miles, ai she performs her revolution about the earth in 27.32166 da; Over how many degrees does she move per hour? Also h( many miles ? TBIOONOMETBIO RATIOS. OB TBIGONOMETBIO FUNCTIONS. Deno 6. Assume any triangle, 0PM, right-angled at M. OM by X, MP by y, and OP by r. Then OP may ho con- sidered as the radius of a circle passing through P and A, and the angle POA, or 0, as being generated by the radius rotating from position OA to OP. A downward rotation of OA would give a negative angle; it being remembered, however, that it is only in tl case of comparison of angles that there arises any necessity f using the terms positive and negative. From (2) the angle O is the ratio « : r. , The other ratios with which we are here concerned are (1) Three leading functions. y - = sine 0, contracted to sin 0, X r y X = cosine 0, n n cos 0, = tangent 0, w u tan 0. miles, and 2166 days. Also how ntio ''. Denote A M •tA ily in the iiessity for ire .A Hi! ilP!i 1 1 III ,„!j ; i m :■ ! :iii TREATISE ON PLANE TRIGONOMETRY. 13 (2) The reciprocals of (1). y ■ cosecant 0, contracted to cosec 0, or cox 0, - = secant 0, X sec 0, cotangent 0, „ „ cot 0. paSuir""!"^ *™^^': "^'^ '"^ "^**'^""' *°^ '^' "^"^^ o^ the The fnl """'* ^ '"'^""y ™^^^^ *°d remembered. Ine following statements may help : the^th! '' •!! ^^"^'"i'^^to' the ratio is sine, cr cosine ; sine when the other side ,s opposite the angle, cosine when adjacent. when tT/.K °""«™^^' -« h«-e «ecant or cosecant; cosecant When the other side is opposite, secant when adjacent. When r does not occur, we have tangent, or cotangent. From the figure of this article, cos <>= - =8in OPSf T ' But 0PM is the complement of 0. Therefore cos <» = sine of complement of .= comp. sine of .^cosine 0, contracted to cos 0. And similarly for other cafunctions. Thus, as 90° and ^ both denote right angles, sin = cos (90° - 0) or cos (^ - 0) ; cot = sin (|- 1 to ^A i>nt AD = x, CD = y. Then y = . tan il = (c - a;) tan if ; whence x is known.) 14. Given two sides of a triangle and the altitude to the thirc side, to find the remaining parts. perpendicu- I 22 inches t.45 inches xlge of the rawn from mgle must o the mid- line makes 20 feet up upon the ad. Find 10 sun is a shadow ° ly, and rts of the ches long, points to ! altitude de to the and the len i/ = x the third ifiil \ i! it I W iipi ! ,. i'\i i 1 \ -J \'\m m m TREATISE ON PLANE TRIGONOMETBY. 17 ». Complement and Supplement. Negative Angle Dmde the circle into '«»Kie. four parts by two orthog- ■ onal diameters, A C and BD. The section AOB {<> the first quadrant, ai 1 any angle lying in t i.s, as -40/*, is > and f>r> -ftOC is the second quad- rant, and any angle lying in this, as AOF, is > 90^ and < 180° Similarly, an angle in the third quad- rant is > 180" and < 270=; and in the fourth quadrant, it is > 270^ and < 360^ .^\ '"S' ^?^ ^' '^'' complement of AOP, and vice versa, smce together they make up the right angle A OB But sini>0^ = :^=^^,,,^^^. I . ?!r^ ^^^ r'"^ '•^ ^° *"e'^ ^« **»« «i"« of its complement I and the sxne of an angle is the cosine of its complement ' Again, ^ AOP 4- pnr ^ ^. . ^^® ^'^' ^•> Then, sin^O/-^^^^,^:^ .^p «m« 0/ a« a Ve is ths same as ih^ ,ine of its (o) Or, th I supplement. X^2 ThlT , n2"'~"* ^'^' "»y """^ «>«- liijiil: liliMj'l ii; ■"\;!'l 18 Now, TREATISE ON PLANE TRIGONOMETRY. ^^ .^p, OAT -OM COS AOP^^^.^—^^^^ _co8 AOP. OF ^P (b) Hence, the cosine of an angle and the cosine of its aujy^ ment are equal in numerical value, but have opposite algebr aigna. Again, an angle of a triangle cannot be greater than 180^ a must always, therefore, be confined to the lirst two quadrkn Angles in the third and fourth quadrants, is those determir by P" and P"', may be considered as taken negatively, in re tion to the angles of a triangle. Thus, AOP" is got by rotati OA backwards in relation to AOP, and AOP" is accordim -AOP, * But the sin AOP"J-^, = Z^^ .^in AOP And .^„„ OM OM cos AOP =.^— „.^_ = cos^OP. (c) Therefore, the sine of a negative angle is minus the sine the equal positive angle ; and the cosine of a negative angle is t same as the cosine of the equal positive angle. Or, when an angle changes sign, the sine of the angle chang sign, and the cosine remains unchanged. EXERCISE VI. 1. Given sin. 21° 35' = 0.36785, to find the sine and tanger of 68° 25'. ^ 2. Given tan. 75°- 3.732, to find the sine and cosine of 15°. 3. Trigonometric tables extend only to 90°. Show how, froi the tables, to find (a) sin. 123°; (6) cos. 165° 44'; (c) tan. 105' 4. Prove that the tangent of an angle and the tangent of it supplement are equal in magnitude and opposite in sign. 5. Show that the limits of magnitude for the sine of an angl are - 1 and + 1 ; that the cosine has the same limits ; and tha the limits of the tangent are from - oo to + oo. 6. Make a table of the variations of the sine, cosine, tangent and secant in each quadrant. ' ila supple- ie algebraic Q 180^ and quadrants, determined ly, in rela- by rotating iccordingly the sine of ngle is the le changes i tangent s of 15°. low, from tan. 105°. ent of its an angle and that tangent, ifl!: ril "1 ,J t| i MM I ""i 1 ,1 I TREATISE ON PLANE TRIGONOMETRY. 19 7. Taking a horizontal linwegnient =6.28 in length to repre- sent the circumference of the circle with radio. 1, oonatract the graphs of the sine, of the cosine, of the tangent, and of the 8. Show from the graphs, or from the table of Ex. 6. that a magnitude changes sign when it passes through .ero or infinity solZj::'^Lr' ""' ''- ^*°^ '- -"^ ''- "■—•''• -^ ^« « curve of THE TBIANOLE. 10. It is shown in geometry that a triangle is given or known when any three of its parts are given, except the three anglls and two sides and the angle opposite the shorter side. When the three angles alone are given, the triangle is given in /o^ colt /h' T^'*"''' ^'"' "'^^ *- «^^- -d thell^e opposite the shorter side are given, there are two triangles,! Z::!:r^'^' the condition, and the triangle is sail tT ^^ The trigonometric solution of triangles consists in finding the remaining parts of a triangle, when three parts, sufficient for longmg to trigonometry come largely under this head. With the use of natural functions the general solution of (a) The sine formula. A£C is a triangle, and £P is the altitude to side CA. Then BPm.a sin C= c sin A, . _«_ ^ c b sin A sin C " sin i? by symm-. ry. And this is the sine for- Imula. 20 ■•iJli! \ i '} i' TREATISE ON PLANE TRIGONOMETRY. Its statement is : In any triangle the sides are proportional the sines of tiie opposite angles. {b) The circumcircle. Let BD be the diameter of the circumcircle of ABC. Then lCDB= lCAB^ i^l, since they stand on the sai arc CB. But sin C2)5=|^=« sin^, or, -^ BD d vm. A < d, and hence a sin A sin B sin C This is, in a way, the completion of the sine formula. EXERCISE VII. 1. In a triangle a = 20, 6 = 26, ^ = 35° 22', to find all tl other parts. 2. In a triangle a -35, 6=48, ^ = 62° 40', to find the oth« parts. Explain the difficulty here. 3. In a triangle A = 51° 20', B~U° 35', c= 17.45, to find th other parts. 4. In a triangle a = 24.60, c = 45.33, j5 = 67° 15', to find th other parts. (Draw CD perpendicular to side c. Then BD ^ a cos 1 and CD = a sin B, and BD and CD are knov.n. Then CD - Aj tan A ; whence ^ ^1 is known. Therefore, etc.) 5. A post 18 feet long leans to the north at an angle of 20 with the vertical. Find the length of its shadow on leve ground, when the sun is south and at an elevation ol 47° 50'. 6. Solve Ex. 5 on condition that the post leans to the eas at the same angle. 7. Solve Ex. 5 on the condition that the post leans to th( sout I at the same angle. 8. A triangle right-angled at J9 has o = 4, c= 10, and the lim BD meets ACin D, and makes the angle CBD - 75°. Find th( length of BD. 9. If in Ex. 8, a- 6, c- 13, at what angle with AB must Bl be drawn, meeting AC in D,9o that BD may be 10? HyrtioncU to the same hence (5) id all the 9 the other B fiud the W find the S = a cos B CD~AD ;leof 20= on level ^° 50'. the east 18 to the the line Find the must BD ( : i •, ! ■ '.. ;ii 'iw -I V, »| \\\\ TBBATISE ON PLANE TRIGONOMETRr. Jl 10. The sides Of a triangle are 13, 14, 15, and the diameter of ite circumcircle u, 16.25. Prove that this is correct, by showing that the sum of its three angles is 180°. "tJ?* ^***^°* Formula.-Thi8 may be developed in seve«J different ways, but the following is one of the simplest. ABU IS a triangle, and BD is the altitude to AC. Theni42)-cco8il/ BD = c sin A. But B(P=BD^-^DC= (c sinii)«+(6-c cosil)». Or, squaring and collecting, a» = i* + c»_2ic cm A. And we have the following sets of forms, which are prac- C tically all the same, being obtained by the principles of symmetry. (o»=J» + c'-26ccos^ 62 = c2 + a^-2(?aco8^ c2 = a« + 6»-2a6cosC' (i) cos A cos 5 = cos C = b^ + e'-a' 2bc c^ + a^-b^ 2m a^ + b'- c^ lab (6) Cor.— If any of the angles concerned are greater than 90° in (o), we must remember that its cosine is negative, Art 9 ^6) and treat it accordingly. * In (6), if the value of the fraction is negative, t e if the thaTgr"" ^ ''*^**'''^' **"'" indicates that the angle is 'greater 22 TREATISE ON PLANE TRIOONOMETllY. ! I I 1 t EXBRCISB VIIl. 1. If a-26, 6-28, c-30, find the angles, sidec^*"*" ""^**^' *-^^-75' ^-27° 15', 5-19° 16'. find t 3. Given «- 1?,71. 6-18.37, A - 14° 47'. i?- 162° 38', to fi, the side c. ' ing pa?tl'" ""*^'^' *"^^'^' '^^ ^-3^°**' *o fi°d *h« w«^ai 5. Starting from ^, I measure off 320 rods in a certain dire won to A I then change my direction through 42° ;,0' an measure off 480 rods to C. How far is it f rom ^ to C in straight line ? 6. The road from ^ to ^ goes by way of C. From ^ to C i l.T^ J^'"*'* '''''^^' *""* from C to J5 is 42 miles 27° east c north. How much will the road from ^ to i5 be shortened b making it direct, and what will be its direction ? 7. Stirting from A, I wish to measure a 10-mile straight line Arriving at ^, 4 miles from A, I find a large swamp. I turn t< the right 50°, and go 2.5 miles. I then go to the left 97° fron my previous course. How far must I go to strike my first lin, at 0, and what distance intervenes between B and C ? 8. In any triangle show that sin A -sin (B+C). 9. In any triangle show that « -6 cos C + c cos B, with twc symmetrical expressions. 10. From Ex. 9 eliminate a, b and c, by means of the sine formula, and show that sin (4 + C) - sin ^ cos C + cos A sin C. 11. The sides of a parallelogram are o and 6, and the angle between t hem is g ; show tha t the two diagonalr are TT ^^"' "*"*'" ^"^ ""^ "^ '^''^ V{a' + b^ + 2ab^^. 11. From Ex. 9 prove the cosine formula by transposing cos C, and squaring. 13. If one diagonal of a parallelogram is double the other, then cos <>- — ('-+*') V, find the J8', to find be remain- tain direc- ° ;.0' and » C in a 4 to C is 7° east of tened by ight line. [ turn to 97° from first line ivith two the sine sin C. le angle isposing 5 other, 1 .1 I I". !■ I i TREATISE ON PLANE TRIGONOMETRT. t8 U. If one diagonal of a parallelogram is a mean proportional between the sides, the other diagonal is V2{a + bY-5ab. 16. The sides of a quadrilateral inscribed in a circle are 4, 7, 5, 8. Find the angles of the figure. 1% The Ambiguous Case.— When an angle of a triangle is determined by its sine alono, it may be either a certain angle or its supplement, since these have the same sine. Thus, if sin il-0.5, then A is either 30° or 150°, and the angle is thus ambiguous. And if there is nothing in the nature of the triangle which excludes one of these values, the triangle is ambiguous, or may have either one of two different forms. An example will make this plain. Ex. 7.— Given a- 50, 6 = 25.87, and A = 30^, to find B. Here sin A ig 87 ■ 0.5 and sin iS = - ^ x 0.5 = 0.2587. 50 ' and ^-15° or 165°. But as the side a opposite the given angle A is greater than the side b opposite B ; .-. ^ A in > . B, and hence 165^ must be rejected, and the triangle is determinate. £x. ^.— Given a- U. 14, 6 = 20, il = 30", to find B -0 . . 20 ^ «n^ = j^^x 0.5 = 0.707, sin B= 14.14' .-. -S = 45°or 135". And as a is < b, either of these angles may belong to the tri- angle, or it is ambiguous, having either one of two forms. 13. Area of the Triangle, in terms of two sides and the mcluded angle. It is shown in geometry that the area of a triangle is one-half the product of the base and altitude. Now, fig. of Art. U,BD = csm A, and the base is b. ^ be sin A . with two symmetrical expressions. (7) -^ i ! B llil inl 'SL'i! 5 I , liill' t \h 'n i \ lli 24 TREATISE ON PLANE TRIOONOMETRy. Dividing a by each number of (7) gives a^_ 2 a 2 whence and eibc 2A „ abc (f -^d (8) gives the circum-diameter or circum-i^ius of the tr angle m terms of its sides and area. Again, 8in»^ » 1 - cos'J = 1 - (^1±^_Z^\ tR\K and factonng the right side as the difference of two squares, w. get «r.»i- (^ + f the tri- uares, we If) '2(«-c), (9) (10) ts three a*; (ex- le sides ides are md 60. }3 feet, : ml if ! ai:i TREATISE ON PLANE TRIGONOMETRY. 25 «d the area is 125 square feet. Find the angle between the sides. Fornt out any ambiguity and explain it. JjJ"" ^"^ ^ '^^ ^"^ ^ ^^^- ^^P^'^ *»»« diffi<"Jty which 7. Given two sides of a triangle, what must be the angle between them that the area may be greatest ? Ja f^ ^^^^^ " *^ ^ ^""""^ ^^ " ^^^^^ o^ 20 feet radius. X::i^'ZZ:' ^ '' '-' -' '^ ^- -^^ *^e othe; eq^i^fz^ il::^?^ f ^*- ^^"^ ''' ^'^ ^^ ^'^ 10. The sides of a triangle are 78. 91. 100. Find the distance from a vertex of a point equidistant from the three vertices. milL . r .* *"*"«^' ^^« «' *' "' fi^d the length of the median m to the side b. wh,Vh -f "^^ ^^l^"^* "^^ ^^^ ™^^' and ^ be the angle and its'L^ni'' ^^ '''' ^'^ ^^«^^ ^^ ^^^^ '^ -^' ^*« --• 13. If ^ be the angle which the long diagonal of the parallelo- gram a. b, 0, makes with side b, show that tan <^- -?L!!?_i_ 6 + a cos ^. ANGLES AS AUXILIAEIBS. JyttolT -^^"^ their functions are often conveniently em- m Whin "^^tri''^'*""* °^ P'^^^^'"^ ^° *-o diff«-nt ways : inte'^lllror^^^^^^^^ principally th«>ugh the S'S'- 1 , li'i 1 1 ll I'l 1 1 ; p. L' ! 1 IPI II ^, -i I" ! i' 26 TREATISE ON PLANE TRIGONOMETRT. As an example of (1), let sin fC'D' + D'E'-[-FA\ But this sum is evidently zero, since we start from A' and en^ at A'. In taking projections analytically we must pay particular attention to the quadrant in which the segment to be projected lies. Thus, AB is in the 4th Q., and as cosines of angles in the 4th Q. are positive, the projection of AB is positive. BC lies in the 1st Q., and its projection is + ; CD lies in the 3rd Q., and its projection is - ; DE, in the 2nd Q, has its projection - j and EA in the 3rd Q. has its projection - . y line m he angle uiy line, C articular rejected the 4th IS in the and its - ; and 'IJJ m V I If. TREATISE ON PLANS TRIGONOMETRY. 29 BXBRCISE XI. 1. Prove by projection that in any triangle b^a cof C + e COS A. 2. In any parallelogram the sum of the projections of two adjacent sides upon the conterminous diagonal is the diagonal 3. ^£CDiB a quadrangle having ^ and C right angles, and the angle ADC iaB: o © », Prove (1) AB sine ^ DC -AD cos e. (2) £C sin 0m AD - DC cos 0. (3) DB sin ff^ViAD'-i-DC'-iAD. DC COB 0). 4. OQ makes angle A with OZ, and OP makes angle A + B with OZ, and PQO is a right angle. Project the triangle OPQ on OX. The sum of the pro- jections of the sides in order is zero. .'. OP cos (A+B) + PQ air. A- QO COS A^O Divide by OP, and But cos (4+^) + ^ sin J -|^ cos ^=0. ^^ ' n ^ QO ^-sin B, and ^-cos B. cos {A + fi) = co8 A cos ^-sin ^1 sin A APPUOATIONS TO FOEOBS AND VELOOITIEa IT. The following fundamental principles are established in the subjects of statics and dynamics. thelnlh "^ r^ ^ completely represented by a line-segment, the length of the segment representing the magnitude of the 80 TBBATISE ON PLANE TRIOOWOMXTBT. Jin. ^^;l force. aj\i\ the direction and position of the segment representin the direr tion and position of the force. (6) TliP acton of a force along any line, or the part of a fore acting in r action parallel to that line, is given by the projec tion up'M Miui line of the segment representing the force. (c) Tr V. u f, -ces M^ represented by adjacent sides of a para] lelogr J ,ey re together exactly equivalent tu the single fore repress iteil !> he conterminous diagonal o{ the parallelogram. Thia s;r .:! ■ 0„oe is allpd the JtenUtant of the two. It i.'i, >a ...>i. ,. , ti, .;, a force exerts no effect in a directioi perper 1 . L. ,,, its own direction, but that it exerts an effeci in every >t her » Ht . tion. The vord "foieo" may be replaced by "velocity" in th« preceding. EXERCISE XII. 1. Two forces of 6 and 15 pounds act at right angles to one another. Find the resultant in pounds, and its directicm rela tively to the greater force. 2. The forces of Ex. 1 act at an angle of 120°. 3. Forces of 30 and 40 grams act at an angle of 55°. Find the force which will exactly annul their effect. 4. Three forces acting at a point are such that their i-epresen- tative line-segments when taken in length and direction can form a triangle. Prove that any one of the forces, reversed in direction, is the resultant of the other two. 5. A ball of weight, W, lies on a plane inclined at angle to the horizontal. Find (a) the force with which the ball tends to roll down the plpne ; (6) the pressure of the ball upon the plane. AB h the plane, making angle (f with Aff, the hori- zoutal, and fT is the ball on the plane. The force in the question is the weight, w, of the ball, and acts vertically downwards. Hence, draw IT^ perpendicular to Aff, to re- present w. treeenting of a force he projec 56. f a paral- igle force ilogram. direction an effect " in the 8 to one Aoa rela- ^ Find •epresen- ;ion can ersed in igle to t«nd8 to e plane. raWHSB OK PLANE -TMOOKOMCTHy. Jj P^«t WJionA£ by the p.,p,„dic„l,r SP. Then Art . 7 C<..-If ..0. the fore, along the pWe « „„, „, ^ P-ur. on the p.«.e i, „. And i, .. g, ,,, ,^ ^ inoLL ^- J'ththo^n"'**" '" '""■"'^ "■« «■• P"- » 7. Interpret the result of 5, when . i, greater than I 8. A car runs up a hill inclined IfiOo* *u . .^ the rope and on the string : ^^ *^* *«°«°° on («) When the puU is horizontal. 10 A t^' P"" " peiT)endicular to the rope tension of the rooe and f K« „ *• , , ™iaaie. To find the on the supports '''^"^^ '^"^ ^°"^*^"*-l forces acting A and ^ are the sup- ports in a horizontal line, and W is the suspended weight. Then evidently each part, AW and BW, of the rope has the same tension and the same in- clination to AB . x-v - V »r- the tension on the rope - 1 82 TREATISE ON PLANE TBIQONOMETBT. The horizontal force on the support is found hy projecting QW on AB~QW cos «--?^ .«;, etc. 2 sin (K 11. The rope of Ex. 10 is 20 feet long and the supports are 16 feet apart. 12. In Ex. 10 the weight is not at the middle, and the inclinations of the two parts of the rope are a and ^ ret>,^- tively. 13. A beam 20 feet long, weighing 5 pounds per foot, stands against an upright wall, and the foot of the beam is 8 feet from the wall. Find the pressure that the beam exerts on the wall. 14. A beam (whose weight may be neglected) has one end on the ground, and the other end is held by a stay rope, so that the beam makes angle a with the horizon and the rope angle /3. A weight, u», is suspended from the upper end of the beam. Find the tension of the stay rope, and the end-thrust on the beam. ADDITION THEOSEM& 18. A theorem which gives a function of the sum or differ- ence of two angles in terms of functions of the separate angles is called an addition theoretn. The principal addition theorems are those for the sine, the cosine and the tangent. These theorems may be developed separately and independ- ently, but, like the functions in general, they can all be derived from any one of them. The addition theorem for cos (A+B) is developed in Ex. 4, of Exercise XI. , We proceed to develop sin (A + B) by another means. The L QOX ^ A and POQ 'B. PQ is ± to OQ, and yr jii.,,^q PJT, ^i/^ are 1 to OX, and QB to PiT. rojecting orts are md the ret>^yec- , stands Jet from ) wall, end on jhat the B/3. A Find }am. : differ- agles is ae, the lepend- lerived Ex. i, i H.\J I i 1 TREATISE ON PLANE TRIGONOMETRY. 33 Then8in(J+^) = :^ sm Now write (A+B) OP^ OP = ^ ^ ^ OQ PQ'OP^ OQOP = cos A. sin B + ain A. cos B. = »in A c(^ B + cos A sia B.. - B for B, and .(«) sin (^-^) = sin.l co8(-i8) + cosil sin ( _ = sin J cos 5 - Ji) Write I" - J for J in sin (A - B), and cos A sin B, (c), Art. 9 (b). sm or (^-^+^)=sin(^^_^)eo8^-cos(|-^) cos(^+^) = cas^ cos^-sin^siniS.!' sin B and finally writing - B for B in the last i^) cos (^ -^) = cos A cos 5 + sin A sin A Collecting the four theorems, we have : sin (A + B)^3mA cos jff + cos ^ sin B sm (A-B)~ sin J cos ^ - cos il sin B cos (^ +^) = cos A cos ^-sin ^ sin ^ cos {A-B)^ cos il cos ^ + sin 4 sin B EXERCISE XIII. (i-) (ii.) (iii.) (iv.). (11) 1. From the addition theorems of (1 1) prove the following • (a) sm 2 = 60°, show that A RAQ~ \,£^RPT (») Find <> when AQ^\aR. Id. Addition Theorem for Tangent- tan lA+B^rrr ^^^ {A + B) ^ 9,1X1 A COS g + c os A sin ^ '"co8(il + if)"cosil cos^-sin A sin 5' Divide numerator and denominator by cos A cos B, and reduce to tangents, and we get Pom the ation u, length > reduce TRfiATISI ON PLANE TBIQONOMXTRT. tan {A + B) tanil + tan B 1 - tan ^ tan ^ and by writing - B for B, and putting tan ( - 5) - tan(^-J5)- **"^-tan^ (16) -tan B, Thenoe we easily obtain tan 2il- 1+tan -J taniS* 2 tan J 1 - tan'J* (17) (18) EXERCISE XIV. 1. Prove that tan (45^ 4- 0) - ^ "*" ^°- ' 1 - tan +./,) sin {li- «^)-(8in tf + sin <^) (sin /?- 8in<^)-8in»tf-8in'''«^. 7. Show that cos {p + ^) cos ()- cos*. 8. Find the sine of 18=. (We have 2 x 18= = the complement of 3 x 18=. and hence sin 2x 18= = cos 3x18=; or 2 sin 18= cos 18= = 4 cos»18°- 3 cos 18=. Divide out cos 18=, and reduce the quotient.) 9. Find the sine anc^ cosine of 3°. (3°- 18° - 16°.) (This is the smalLst whole number of degrees of which we can find the functions in terms of surd expressions. Thence, they can be so found for every three degrees throughout the quadrant. 86 TRIATISI ON PLANE TBIOONOMKTRY. SO. FormuUs for changing Sums and Differences of Functions to Products. Add (i.) and (ii.) of (11), Art. 18, and we get sin (J +^) + 8in {A~B)'^2 ain A cm B. Now pnt A" ^(), Bm, -{0- «/>), and, therefore, A-k-B^O, and A - B^ , and we have sin + nn ). Similarly, by subtracting (ii.) from (i.), by adding (iii.) to (iv.), and by subtracting (iv.) from (iii.), we get the four forms : sin 6^ + sin ^ « 2 sin -^{0 + tp) cos -{0-^) sin /> - sin 9 - 2 cos ^(tf + f) sin ^{0 - i 2 cos 8sess TREATISE OW PLAXt TRIOONOBIETRT. (a) Prove that t«n ^ + Un 5 + tan C » tan il tan ^ Un C. (/») cot J cot ^ + cot £ cot C + cot C cot -4 - 1. (c) sin ^ +8in i? + sin C- 4 cos ^ COM - cos -. 2 2 2 (»in ^ + gin jB- 2 sin ^-ti cos ^~^, and 87 c c sin C- 2 sin ^ cos ^ = 2 am *^V2— 2-^-2 (2-^) A+B . A-B cos — -— sin /. sin i4 +8in B + mi C= 2 cos ;• (coft — -- 4. cos 2 A- fi A-¥B r^) .ABC «4 cos - cjw cos ; .). LOGABITHUIO FOBMULAS. SJI. Logarithms are employed to simplify and extend arith- metical operations, and their proper relation is with algebra and arithmetic. They are introduced into practical trigonometry not as a matter of necessity, but as one of convenience, aud Ijecause a large part of the work in that subject consists of arithmetical operations. The trigonometrical functions, being ratios, are numbers, and the logarithms of these numbers are tabulated under the head of log-sines, log-tangents, etc. The tabulating of them in this way is a great convenience, but all these logarithms can, of course, be got from a table of logarithms of numbers. The log-sines, etc., offer some peculiarities, for the natural quantities being fractional, their logs have a negative charac- teristic ; and to get over this inconvenience 10 is added to the characteristic. Rules for working these tables are generally found in conjunc- 38 TBBATISE ON PLANI TRIOONOMTnBT. tion with the tables, and there is no advantage in giving them here, as facility in the use of the tables is to be acquired only by practice and experience. We shall, therefore, assume that the reader is acquainted with the general properties of logarithms, and that he has some knowledge of the tables. For convenience we here state the three most important working properties of logarithms : (a) log a + log 6 - log ab. a (b) loga-log6-l(w T. o (c) n log a - log o" for all values of n. Thus, as the addition of logarithms corresponds to the multi- plication of numbers, and the subtraction of logarithms to the division of numbers, there is no operation with logarithms corresponding to the addition or subtraction of numbers. And Ijeing given log. o + log. ft, there is no direct logai-ithmic means of finding log. (a + ft), except by repeated operations. Hence, formulas involving additions or subtractions are not adapted to logarithms, and when additions or subtractions are necessaiy, they must be effected before the application of logarithms. Thus the sine formula is adapted to logarithms directly, since it involves only multiplications and divisions. But the cosine formula is not adapted to logarithms, as it involves additions and subtractions of the squares of quantities ; and if these arifh metical operations are to be carried out first, the subsequent application of logarithms would be more laborious than helpful Hence the nec^sity of transforming our formulas, so as to adapt them to logarithmic computation. tX. Transformation of Cosine Formula. (a) cos A b^ + c--d' 2be 2 cos='^=l+ cos .1. !+*' + '''-«' 26c (b + cY-a ' ■2bc I Jill TREATISS ON PLAJn TRIOONCMETRY. 89 COS COS , ^ _ (a + 6 + c)(6 + c -a) *(« - a) 4be 6c (20) This is adapted to lofwithms, since the only additioiu ud subtractions are between simple numbers, and not between squares of numbers or trigonometric functioac Its logarithmic form is /. cos - - ^ {/.* + t.{s-a). Lb - U}, (20/.) and this serves to find an angle when the three sides are given. (b) 2 sm- A • 1 - cos il -1 1 - ■2bc ' 2bc and sin sin A (a + f>-c)(a- b + r) (g-bXs-c) 2 N/" be 46c be The logarithmic form is '• «^ \ = 72 {'•(« - *) + '•(» - c) - l.b - Lc}, and this serves the same purpose as (a). (c) Another method is by an auxiliary angle. (21) (21/.) As in (a) and cos--- A (b + cy-d' 2bc 2 46r V \b + cJ )* A^ b + c 1 / a'V' Now, as o, 6, c are sides of a triangle, b + c > a, therefore a . h':^c ^"^ '®** ****" ^^itj, is the sine of some angle, say. I? 40 TREATISE ON PLANE TRIGONOMETRY. Then. COS A 6 + . 2 2Vbc — CW0, and we have the two logarithmic forms ; (1) /. sin tf-/.o -/.(& + e). {2)1 1 COS 2 - l{b + c) + Z. cos 5. In the triangle ABC, « = 27.3, 6-34.1, o = 45.6, to find the angle A. («-=53.5, «-ae26 2. Then A 1 /. cos -^ = 2('-^^-5 + '-26.2-/.34.1 -/.45.6) = 9.97747. •• 2 = IS" 18, etc. 4< TBEATI81 ON PLANE TRIQONOMITRV. WVBMB. OB OntOiyLAB PUHOTIOHB. ^^LT^ r ^*7 "; '"*' ""^ "'•y J^f ^-dn-'x, which dLl "^.^^-^ «ne of X. The exponent, - 1. here do^ not denote a reciprocal, as in algebra. fn^^ ^""^V"^ ''^'^^ «!"»» in number the trigonometric ir-rei."" " '^'^" ^"'''"^' "" "^■^' ^■'-. the^'Sff^ T- \'*^«'?l^°'''°'" *" ^"PO'^^* i« the subjects of ^J^!^7 u '^^ ^''*^^ ^•^^"^"^ The important theorems m regard to them are, however, not numerous (a) To sum tan-'x + tan-'y, that is, to find tangent equal to the sum of these. an inverse Let Then and or Or, finaUy, - tan-'a and » tan-'y. tan « a; and tan 0- y. fatn(^ + ff)_ tan»+tang 1 - tan tan (/' + fl - tan tan-'x + tan-'y « tan"' -i«+y and similarly, tan-'x - tan"' '!/' \-xy \-xy" X-'i 1 +ary ' tan-'- — — (16) (24) (26) ^a:.— tan-'- + tan"'- « tan-' 1 1 2-^3 2 3 ■ tan-'l .45° or ;. (b) To find the sum of sia-'a: + sin-'y. Let Then » sin '« and — sin-'y. sin^BX, sin (9 = y. cos - Vl -3?, cos fl » Vn^, s tl n 01 TBBAnsI OK PLANI TRIOONOMITBY. »nd tin (f + 0) » sin » cog, fl + cos ^ sin e. 8in-'x + sin-y - sin-' {* VPTp + y VIT^} . BXBRCISB XVII. 1. Prove that sin-'a? - - - co8-'«. mi 2. Find 2 tan-»- as an inverse tangent. 3. Show that 2 tan-»l+2tan-i +tan-'1.45^ 4. Prove that 4 tan-' J - ten-'— -45°. 5 239 5. Prove that "sin-'x + Hin-y = cos" { y/\:i^>/TI7f - xy } t>. J? md a circular function equal to cos"'* + cos-'y. 7. Show that 8in-'| + sin"'* - "'. 8. Show that 2 tan ^ + tan-'- . \fP 3 7 48 fii m TEIOONOMETEIOAL OONSTBUOTIONS. «6. By trigonometrical construction we mean the finding, by i^raphical methods, of the values of such trigonometrical expres «ons as can be so found, and which have sufficient elements given to make them determinate. On account of the great variety of such expressions, no very general principles of construction can be laid down, and even the constniction for a given case may sometimes admit of a number of variations, of which some are more elegant than A few examples will illustrate the subject. MHCROOOrV RBOUniON TBT CHART (ANSI and ISO TEST CHART No. 2) A /APPLIED IM/^GE I, nc 1653 Eatt Main StrMt RoctwtUr. New Yrr U609 USA (716) ♦«2-0yi-Phon« (716) 288 - S9.'I9 - Fax 44 TREATISE ON PLANE TRIGONOMETRY. Ex. /.—To construct an angle when its sine is given. Take any line-segment, OP, as a radius, as the element of length must be involved, and on it describe the semicircle OMP. In this semi-circle place the chord PM equal to OP X the given sine, and join OJf. The ^ POM is that required. The proof is evident from the construction. Ex. ^— To construct a sin 6 cos H, where a and B are given Draw a line OP = a, and OM making lPOM=0. Draw PM ± OM, and MJfT ± OP. Then OM=a cos d, and MN=^OM sin 61. . • . MN= a sin cos B. ^ Ex. 3. — To construct sin 4- cos 61 a. — , or a(sin B + cos B) cot B BAD^B, and tan B Draw AB = a, and make draw BC ± AD. Take CD = CB, and draw AE 1 AB and DE 1 AD, to meet in E. Then, ^(7 = o cos 0and Ci? = rt sin. 6, and .-. i42) = a(8in 0-|-cos B). But lAED = B, and DE^AD cot ^^Z), . •. />^ = a(8in + cos ft) cot 0. TREATISE ON PLANE TKIGONOMETRY, 45 EXERCISE XVIII. 1. Construct an angle when (a) its tangent is given ; (6) when its secant is given ; (c) when its cotangent is given. 2. Construct x where x =. (sin A - sin Ji) V^a- -T', where A, B are given angles, and a, h are sides of a given rectangle. 3. Find by construction the rectangle ab sin ^cosf, where a, b are given line segments and ^ is a given angle. 4. Construct B where « = sin-'^, a and b being given line segments. 5. Construct tan"' _ + tan"' - . 6. Construct a sin ('^ + tan-»-j, where A is a given angle anf' rt anJ 6 are given line segments. 7. Conotruct the graph of sin f/ - cos 6/ from 0= to fl= 2;r. 8. Construct the graph of sin 6/ + cos ^ from to 2;r. 4(] TREATISE ON PLANE TRIGONOMETRY. MISCELLANEOUS EXERCISES. 1. C is the centre of a circle with radius r, and P is a point without. Tangents from P touch the circle at T and T'. Find the tangent of TPT in terms of r and PT. 2. In Ex. 1 find the length of chord T7" 3. The distance between graduation marks on the limb of a theodolite is 0.045 inch, and they represent 10' of angle. Find the radius of the limb. 4. Prove the following relations : (a) cosM - sinM = 2 cos^.4 - 1. (b) Vl - sin 6^ = (sec - tan ^)Vl +sin (f. cosec $ cosec $ 2 secV = cos A cosec (t- I cosec ^+r sin A = sin A +COS A. 1 - tan A 1 - cot A (e) 4 cos'.4 = 3 cos il + cos 3 A. (/) sec* A - se 5 U = tan* A + tan- A. 5. Find any function of from the following : (a) 2 sin © = 2 -cos $. (b) 8 sin 6^ = 4 + cos ft. (c) tan d + sec 6 = 3/2. {d) sin H + 2 cos ^=1. (e) tan2fl + cot0 = 8cos2ft (Express left-hand member in sines and cosines and reduce.) 6. In any cirMe prove that the chord of 108° is equal to the fium of the chords of 36° and 60°. 7. A person standing on a lighthouse notices that the anjle of depression of a boat coming towards him is «, and that after m minutes it is §. How long after the first observation will it be before the boat reaches the lighthouse ? TREATISE ON PLANE TKIOONOMETKV. 8. (a) From the cosine formula shov that 47 c^{a + b) sm ^ 8ec0, where . a-b C tan if, = -—- - cot apply htfthr *^\''"''' "^'^''^ '" '^«'^"*^'"- f-™> and apply It to the case where a = 25.33, 6= 18.46, and C-78= 44' 9. («) Prove that aco^e + bsinO^ VcT-H? cos U - tan- * ) 'Va^ + 6-'8in(tf + tan-'-^]. (A) Show that a cos ^ + 6 sin 6 is a maximum when^ »-tan-'-. of 2 i^n- ^'"f ?' ^"^^' ^ ^"''^ ""^ P'^'-*^' «"«'» that the sum of the cosmes of the parts is a given quantitv, ,«. {b) Obtain a geometric construction for this division. 11. Prove that tan->- - tan->^^^*= "^ " m + n 4' 1- A ^ide of a triangle is 4 and the opposite angle is 36^ and the altitude to auother side is V5. 1. Find the other parts. into '^ ' . "f ''" *" '''^' " '' "^' «^d the parts ihe tr. : "' ""^^' '^^ "' ^- ^'"^ *h^ -»her par^ of 14^ Soive the triangle in which a + 4, c and C are given (Find a-bhy cosine formula.) ^ "' 15. The altitude of a rock i'« 47° a**^ n • ... '"*''' IS 4/ . After walkmc 1 000 f^^f tical height of the rock above the .first point of observation. lb. (a) A hill which rises 1 in 5 faces south. Show that a road on It which takes a N.-E. direction rises 1 in 7 (6) What must be the direction of the road which goin^ along the hill, rises 1 in 10? ^ ^ th.V' t ^^^u^ ^^"'''^ ''''''^ ^"^ * ^^'-^^^^^ '^"gle of 2v. Show that when the sun is south at elevation «, the angle of thi shadow of the gable on level ground is 2 tan - (tan « tan ,) 4» TREATISE ON PLANE TRIGONOMETRY. 18. A rod points towards the north pole of the heavens. Find the angle, 6, made by the shadow of the rod (on the level) with the meridian line, when the sun is h degrees west of the merid- ian, and the altitude of the pole is ^. (This emlxxlies the principles of construction of a horizontal sun-dial.) 19. (rt) In Ex. 17 give a geometric construction for finding ^ when h and are ;^ven. (6) Lay oflf the hours of a dial for latitude 44=' N. 20. (a) The quadrilateral OPMQ, in order, has OP^OQ = r, the L P0Q = 2y, and the angles OPN tuad OQM equal to u and fl respectively. Find an expression for the value of PM. (This embodies the principle of finding the distance ..t the m(X)n from the earth. ) (6) If in (a) «= 145^ |i* 164= 12', and y = 25^, show that yj/— 60>* very nearly. 21. (a) ABCD, taken in order, determine a trapezoid with .(15 parallel to 2>C; and the diagonals meet in 0. The angle DAC^a, ADB=2py and the ratio xiC:OC = r. Show that /> = (r- 1)- , if the angles ^> and « be very sm.ill. (b) Find/, when «-46''.8 and /•= (^^^-^V 22. If r^ be the radius of the circle escribed to side a of the B C triangle ABC, show that r' = a a cos - cos _^ B C tan - + tan - cos A 23. Prove the following : {'t) rirjr3 = 8^ = r»^. (b) rr^r^r3 = ^\ (c) at2R sin A, where R is the circumradius. ((/) r, = 4^ sin — cos — cos -, with symmetrical expres- sions for r., and r^. e) r-4^sin— sin — sin -. il? TREATISE ON PLANE TRICJONOMETRV. 49 24. Two wlieel.. with rae{8in C + sin C cos.C / Vn^-sin^C}, where n is the r&tio A B.BC. (This exercise embodies the principles of the cross-head and crank in the steam engine. ) 34. From the comer of a cuboid a piece is cut oflf by a plane saw cut, which reaches to the distances a, b, c respec- tively on the three edges. Prove that the area of the section is IVa'b' + b-'c' + c'a^ 35. At the vertices of an equilateral triangle line segments, a, b, c respectively, are drawn normal to the plane of the triangle. Show that the area of the triangle formed by connecting the outer points is -V {3f>* + ig'iSa' - 2ab)}, where s is the side of the equilateral triangle. TREATISE ON PLANE TRIGONOMETRY. 51 ANSWERS TO EXERCISES. 1. 3. 6. 9. 11. Exercise I. 16581608928. 2. 219.608928 ; 3.128928 ; 0.039072 ft. 3.U1593. 4. 4; 5. 5. 5 • 7 3.178 sq. ft. 7. 1.0000000. 8. 1.00000. 326 sq. ft. 60 sq. in. lo. 0.062 0.4342945. 12. 57.295780. 13. 101.881. 1. 36^.2389. 4. 3° 50' 49".92. 7. 60=. 10. 49' 8". 13. 1082.3 ft. 16. 3977 m. 18. 0^.549; 2270.96 m. Exercise II. 2. 0°. 42093. 5. 0.30718 in. 8. 0r8864. 11. 214.9 ft. 14. 15.079 ft. 17. 18.52 m. 3. 0°.015097. 6. ".0098 15. 9. 55.178 ft. 12. 412.5 ft. 15. 345.6 m. 3. |V2-; \V2. -345 5' 5' 4" Exercise IV. 8. (i-«^)^rrp,, Exercise V. 1. 15= 54' 56"; 39.52 in. 0. ,. 47, 3 ^g^ 3^, ' 5. 18=26'; 71° 34' 6. («) 96.84 ft; (6) 70.87 ft. 7.17.76 ft 8 27° 49' 9. C-7r 29'; a = 34.41 ; 6 = 26.63. 10 2 495 in 11. 5P32'; 59=52': 68=36' 13. Cr = 84=58';a-17.1.56; 6=11.896. 52 TREATISE ON PLANE TRIGONOMETRY. Exercise VI. 1. 0.92988; 2.52786. 2. 0.259; 0.966. EXEBCISR YII. 1. ^=48° 48': C-95°50'; c = 34.37. 3. C-112°5'; a -14.70; 6-6.37. 4. 6-42.42; J -32° 21' 12"; C-80=23'48". 5. 21.47 ft. 6. 16.5 ft. 7. 9.16 ft. 8. 6.2. 9. 8° 15'. Exercise VIII. I. A^ 53° 8' ; 5 = 59^ 32' ; C- 67= 20'. 2. 38.50. 3- 2.79. 4. c- 34.38; .4 = 48=52'; .fi-93°24'. 5. 747 rods. 6. 1.64 m.; 17= 31' east of N. 7. 2.62 m. ; 3.39 m. 15. ^^0= 10' ; 92° 34' ; 79° 50' ; 87= 26'. , 4 56 12 "' 5' 65' 13* 5. 51° or 129= 9. 108 25 ft. Exercise IX. 3. 156.4 rods. 4. 1344. 8. 26.25 ft. ; 23= 35'; 17= 27' : 138° 58' 10. 52.5. 11. 7iV2c'' + 2a' -b-. Exercise X. 3. {Vim' +1-1) I 2m. \\tt3' 10. 2.801. Exercise XII. 1. 16.16 lbs. ; 21° 49' with greater force. 2. 13.08 lbs. ; 23° 25' with greater force. 3. 62.26 grms. ; 156° 45' with greater force. 6. 51.96 lbs. ; 30 lbs. 4J] TREATISE ON PLANE TRIGONOMETRY. S.j q" f f ^* """^ ^^ ho"*- ; 1 236 miles per hour. 9. («) 100.38 lbs. ; (A) 99.62 lbs.. 8.72 lbs. 11. Tension. 1^.. horizontal force- ^ vertical force. I,. 1 I) . to cos 3 in cos « i. ^j--: — . £-. /.,«-__ri^!" U^, « to cos « COS d sm (a + d)' * sin 'irrT»» no*^- force-—; !^ 13. 21.82 iL ' "' "■""^^ Exercise XIII. 5. (V/3-1) 2v/2T(l/3+l)/2V'2' 7. (a) 45^(.)^.l^,^,^e0^^(.) 60^,(0 530 8'. „ 3 tan fl - tan'^ i ,_ • "TT3Ta-iii^- 8.4(^5-1). Exercise XIV. ., \/3-l V3"+l ■■ V3+1' V3'_r 9. {( V£ji)(v/3 + 1 ) - VToT-ivMVs- 1)} / sV^J {V/l0 + 2V5(v/3 + l) + (V5-l)(V3-l)}/Sv/2-: Exercise XV. 1- - (cos 0-oOS 3m. sin /J^P\ * ■* ' — »'Q ("^ + xf) ; cos A cos ^. Exercise XVI. 1. (a) 16^ 23' 40"; (6) 1= 4I' 51"; (c) 2= 6'- (d) 15' 58' 2*5". («) 89' 46' 48" 'WO, W ID 58 25 ; 111 if i^XERCISE XVII, 2. tan-'-. 4 6. cos-ilary-Vl-ar! VT ■yi. -««fc''*-'*i 54 TREATISE ON PLANE TRIGONOMETRY. M18CULAKBOU8 EXBKCISBS. 1. 2r.PT/ {PT'-r'). 3. 16.47 in. .m A 2. 2r.PT I VP2'' + r*. m COS a sin £< sin (p{ - a) 10. (a) 2^-J + cos->'"8ec ' 12. (V6 - 1) cosec 36°, and {y/b - 1) cot 36* + A^ioTivI! 13. 2m sin a / sin (a + ^), 2»» sin ^ / sin (a + pl). 15. 1000V2 8in47' 16. (6) N.E. 15' S., or N.W. 15' S. 18. tan 0«8in ^ tan A. 24. (a) 2 - (»• + »•> - (r + r') cos-'^^' + V(/='-(r+r')». (b) -^r7r-(r- r') cos-'-lI + ^d' - (r - /)«. 30. 44.8 kgms. 31. (a) 69' 44'; (6) 18.76 ft. 32. (a) 35.5 lbs.; (b) 12.31 lbs. ftgtmmmmtmmntmamm