Trigonometry - Bayanbox.ir
Trigonometry
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TrigonometryWith Calculator-Based SolutionsFourth EditionRobert E. Moyer, PhDAssociate Professor of MathematicsSouthwest Minnesota State UniversityFrank Ayres, Jr., PhDFormer Professor and Head, Department of MathematicsDickinson CollegeSchaum’s Outline SeriesNew York Chicago San FranciscoLisbon London Madrid Mexico CityMilan New Delhi San JuanSeoul Singapore Sydney Toronto
Copyright 2009, 1999, 1990, 1954 by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United StatesCopyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrievalsystem, without the prior written permission of the publisher.ISBN: 978-0-07-154351-4MHID: 0-07-154351-1The material in this eBook also appears in the print version of this title: ISBN: 978-0-07-154350-7, MHID: 0-07-154350-3.All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked name, weuse names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark. Where suchdesignations appear in this book, they have been printed with initial caps.McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs.To contact a representative please visit the Contact Us page at www.mhprofessional.com.TERMS OF USEThis is a copyrighted work and The McGraw-Hill Companies, Inc. (“McGraw-Hill”) and its licensors reserve all rights in and to the work. Use ofthis work is subject to these terms. Except as permitted under the Copyright Act of 1976 and the right to store and retrieve one copy of the work,you may not decompile, disassemble, reverse engineer, reproduce, modify, create derivative works based upon, transmit, distribute, disseminate, sell,publish or sublicense the work or any part of it without McGraw-Hill’s prior consent. You may use the work for your own noncommercial andpersonal use; any other use of the work is strictly prohibited. Your right to use the work may be terminated if you fail to comply with these terms.THE WORK IS PROVIDED “AS IS.” McGRAW-HILL AND ITS LICENSORS MAKE NO GUARANTEES OR WARRANTIES AS TO THEACCURACY, ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE OBTAINED FROM USING THE WORK, INCLUDING ANYINFORMATION THAT CAN BE ACCESSED THROUGH THE WORK VIA HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIMANY WARRANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO IMPLIED WARRANTIES OF MERCHANTABILITY ORFITNESS FOR A PARTICULAR PURPOSE. McGraw-Hill and its licensors do not warrant or guarantee that the functions contained in the workwill meet your requirements or that its operation will be uninterrupted or error free. Neither McGraw-Hill nor its licensors shall be liable to you oranyone else for any inaccuracy, error or omission, regardless of cause, in the work or for any damages resulting therefrom. McGraw-Hill has noresponsibility for the content of any information accessed through the work. Under no circumstances shall McGraw-Hill and/or its licensors be liablefor any indirect, incidental, special, punitive, consequential or similar damages that result from the use of or inability to use the work, even if any ofthem has been advised of the possibility of such damages. This limitation of liability shall apply to any claim or cause whatsoever whether suchclaim or cause arises in contract, tort or otherwise.
PrefaceIn revising the third edition, the strengths of the earlier editions were retained while reflecting changes inthe vocabulary and calculator emphasis in trigonometry over the past decade. However, the use of tablesand the inclusion of trigonometric tables were continued to allow the text to be used with or withoutcalculators. The text remains flexible enough to be used as a primary text for trigonometry, a supplementto a standard trigonometry text, or as a reference or review text for an individual student.The book is complete in itself and can be used equally well by those who are studying trigonometry forthe first time and those who are reviewing the fundamental principles and procedures of trigonometry.Each chapter contains a summary of the necessary definitions and theorems followed by a solved set ofproblems. These solved problems include the proofs of the theorems and the derivation of formulas. Thechapters end with a set of supplementary problems with their answers.Triangle solution problems, trigonometric identities, and trigonometric equations require a knowledge ofelementary algebra. The problems have been carefully selected and their solutions have been spelled out indetail and arranged to illustrate clearly the algebraic processes involved as well as the use of the basic trigonometric relations.ROBERT E. MOYERv
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ContentsCHAPTER 1Angles and Applications11.1 Introduction 1.2 Plane Angle 1.3 Measures of Angles 1.4 Arc Length1.5 Lengths of Arcs on a Unit Circle 1.6 Area of a Sector 1.7 Linear andAngular VelocityCHAPTER 2Trigonometric Functions of a General Angle102.1 Coordinates on a Line 2.2 Coordinates in a Plane 2.3 Angles in StandardPosition 2.4 Trigonometric Functions of a General Angle 2.5 Quadrant Signsof the Functions 2.6 Trigonometric Functions of Quadrantal Angles2.7 Undefined Trigonometric Functions 2.8 Coordinates of Points on a UnitCircle 2.9 Circular FunctionsCHAPTER 3Trigonometric Functions of an Acute Angle263.1 Trigonometric Functions of an Acute Angle 3.2 Trigonometric Functionsof Complementary Angles 3.3 Trigonometric Functions of 30 , 45 , and 60 3.4 Trigonometric Function Values 3.5 Accuracy of Results Using Approximations 3.6 Selecting the Function in Problem Solving 3.7 Angles of Depressionand ElevationCHAPTER 4Solution of Right Triangles394.1 Introduction 4.2 Four-Place Tables of Trigonometric Functions 4.3 Tablesof Values for Trigonometric Functions 4.4 Using Tables to Find an Angle Givena Function Value 4.5 Calculator Values of Trigonometric Functions 4.6 Findan Angle Given a Function Value Using a Calculator 4.7 Accuracy in ComputedResultsCHAPTER 5Practical Applications535.1 Bearing 5.2 Vectors 5.3 Vector Addition 5.4 Components of a Vector5.5 Air Navigation 5.6 Inclined PlaneCHAPTER 6Reduction to Functions of Positive Acute Angles666.1 Coterminal Angles 6.2 Functions of a Negative Angle 6.3 ReferenceAngles 6.4 Angles with a Given Function ValueCHAPTER 7Variations and Graphs of the Trigonometric Functions747.1 Line Representations of Trigonometric Functions 7.2 Variations ofTrigonometric Functions 7.3 Graphs of Trigonometric Functions 7.4 Horizontaland Vertical Shifts 7.5 Periodic Functions 7.6 Sine Curvesvii
viiiCHAPTER 8ContentsBasic Relationships and Identities868.1 Basic Relationships 8.2 Simplification of Trigonometric Expressions8.3 Trigonometric IdentitiesCHAPTER 9Trigonometric Functions of Two Angles949.1 Addition Formulas 9.2 Subtraction Formulas 9.3 Double-Angle Formulas9.4 Half-Angle FormulasCHAPTER 10 Sum, Difference, and Product Formulas10610.1 Products of Sines and Cosines 10.2 Sum and Difference of Sines andCosinesCHAPTER 11 Oblique Triangles11011.1 Oblique Triangles 11.2 Law of Sines 11.3 Law of Cosines 11.4 Solutionof Oblique TrianglesCHAPTER 12 Area of a Triangle12812.1 Area of a Triangle 12.2 Area FormulasCHAPTER 13 Inverses of Trigonometric Functions13813.1 Inverse Trigonometric Relations 13.2 Graphs of the Inverse TrigonometricRelations 13.3 Inverse Trigonometric Functions 13.4 Principal-Value Range13.5 General Values of Inverse Trigonometric RelationsCHAPTER 14 Trigonometric Equations14714.1 Trigonometric Equations 14.2 Solving Trigonometric EquationsCHAPTER 15 Complex Numbers15615.1 Imaginary Numbers 15.2 Complex Numbers 15.3 Algebraic Operations15.4 Graphic Representation of Complex Numbers 15.5 Graphic Representation of Addition and Subtraction 15.6 Polar or Trigonometric Form of ComplexNumbers 15.7 Multiplication and Division in Polar Form 15.8 De Moivre’sTheorem 15.9 Roots of Complex NumbersAPPENDIX 1 Geometry168A1.1 Introduction A1.2 Angles A1.3 Lines A1.4 TrianglesA1.5 Polygons A1.6 CirclesAPPENDIX 2 Tables173Table 1 Trigonometric Functions—Angle in 10-Minute IntervalsTable 2 Trigonometric Functions—Angle in Tenth of Degree IntervalsTable 3 Trigonometric Functions—Angle in Hundredth of Radian IntervalsINDEX199
Trigonometry
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CHAPTER 1Angles and Applications1.1 IntroductionTrigonometry is the branch of mathematics concerned with the measurement of the parts, sides, and anglesof a triangle. Plane trigonometry, which is the topic of this book, is restricted to triangles lying in a plane.Trigonometry is based on certain ratios, called trigonometric functions, to be defined in the next chapter.The early applications of the trigonometric functions were to surveying, navigation, and engineering. Thesefunctions also play an important role in the study of all sorts of vibratory phenomena—sound, light, electricity, etc. As a consequence, a considerable portion of the subject matter is concerned with a study of the properties of and relations among the trigonometric functions.1.2 Plane AngleThe plane angle XOP, Fig. 1.1, is formed by the two rays OX and OP. The point O is called the vertex andthe half lines are called the sides of the angle.Fig. 1.1More often, a plane angle is thought of as being generated by revolvinga ray (in a plane) from the initialSSposition OX to a terminal position OP. Then O is again the vertex, OX is called the initial side, and OP iscalled the terminal side of the angle.An angle generated in this manner is called positive if the direction of rotation (indicated by a curved arrow)is counterclockwise and negative if the direction of rotation is clockwise. The angle is positive in Fig. 1.2(a)and (c) and negative in Fig. 1.2(b).Fig. 1.21
2CHAPTER 1 Angles and Applications1.3 Measures of AnglesWhen an arc of a circle is in the interior of an angle of the circle and the arc joins the points of intersectionof the sides of the angle and the circle, the arc is said to subtend the angle.A degree ( ) is defined as the measure of the central angle subtended by an arc of a circle equal to 1/360of the circumference of the circle.A minute ( ) is 1/60 of a degree; a second ( ) is 1/60 of a minute, or 1/3600 of a degree.1EXAMPLE 1.1 (a) 4s36 24rd 9 6r11(b) 2s127 24rd 2s126 84rd 63 42r11(c) 2s81 15rd 2s80 75rd 40 37.5r or 40 37r30s111(d) 4s74 29r20sd 4s72 149r20sd 4s72 148r80sd 18 37r20sWhen changing angles in decimals to minutes and seconds, the general rule is that angles in tenths willbe changed to the nearest minute and all other angles will be rounded to the nearest hundredth and thenchanged to the nearest second. When changing angles in minutes and seconds to decimals, the results in minutes are rounded to tenths and angles in seconds have the results rounded to hundredths.EXAMPLE 1.2 (a) 62.4 62 0.4(60 ) 62 24 (b) 23.9 23 0.9(60 ) 23 54 (c) 29.23 29 0.23(60 ) 29 13.8 29 13 0.8(60 ) 29 13 48 (d) 37.47 37 0.47(60 ) 37 28.2 37 28 0.2(60 ) 37 28 12 (e) 78 17 78 17 /60 78.28333. . . 78.3 (rounded to tenths)(f) 58 22 16 58 22 /60 16 /3600 58.37111. . . 58.37 (rounded to hundredths)A radian (rad) is defined as the measure of the central angle subtended by an arc of a circle equal to theradius of the circle. (See Fig. 1.3.)Fig. 1.3The circumference of a circle 2 (radius) and subtends an angle of 360 . Then 2 radians 360 ; therefore180 1 radian p 57.296 57 17r45sandwhere 3.14159.1 degree pradian 0.017453 rad180
3CHAPTER 1 Angles and ApplicationsEXAMPLE 1.3 (a)7p # 180 7p rad p 105 1212p5p(b) 50 50 #rad rad18018p 180 p(c) rad # p 30 66p7p(d) 210 210 #rad rad1806(See Probs. 1.1 and 1.2.)1.4 Arc LengthOn a circle of radius r, a central angle of radians, Fig. 1.4, intercepts an arc of lengths ruthat is, arc length radius central angle in radians.(NOTE: s and r may be measured in any convenient unit of length, but they must be expressed in the same unit.)Fig. 1.41EXAMPLE 1.4 (a) On a circle of radius 30 in, the length of the arc intercepted by a central angle of 3 rad is1s r u 30 A 3 B 10 in(b) On the same circle a central angle of 50 intercepts an arc of lengths r u 30a5p25pb in183(c) On the same circle an arc of length 112 ft subtends a central angle183s radu r 305or3 2s3u r rad55 2when s and r are expressed in incheswhen s and r are expressed in feet(See Probs. 1.3–1.8.)
4CHAPTER 1 Angles and Applications1.5 Lengths of Arcs on a Unit CircleThe correspondence between points on a real number line and the points on a unit circle, x2 y2 1, withits center at the origin is shown in Fig. 1.5.Fig. 1.5The zero (0) on the number line is matched with the point (1, 0) as shown in Fig. 1.5(a). The positive realnumbers are wrapped around the circle in a counterclockwise direction, Fig. 1.5(b), and the negative real numbers are wrapped around the circle in a clockwise direction, Fig. 1.5(c). Every point on the unit circle is matchedwith many real numbers, both positive and negative.The radius of a unit circle has length 1. Therefore, the circumference of the circle, given by 2 r, is2 . The distance halfway around is and the distance 1/4 the way around is /2. Each positive number is paired with the length of an arc s, and since s r 1 . , each real number is paired withan angle in radian measure. Likewise, each negative real number is paired with the negative of thelength of an arc and, therefore, with a negative angle in radian measure. Figure 1.6(a) shows points corresponding to positive angles, and Fig. 1.6(b) shows points corresponding to negative angles.Fig. 1.6
5CHAPTER 1 Angles and Applications1.6 Area of a SectorThe area K of a sector of a circle (such as the shaded part of Fig. 1.7) with radius r and central angle radians isK 12r2uthat is, the area of a sector 12 the radius the radius the central angle in radians.(NOTE: K will be measured in the square unit of area that corresponds to the length unit used to measure r.)Fig. 1.7EXAMPLE 1.5For a circle of radius 30 in, the area of a sector intercepted by a central angle of 13 rad is111K 2 r2u 2(30)2 A 3 B 150 in2EXAMPLE 1.6For a circle of radius 18 cm, the area of a sector intercepted by a central angle of 50 isK 12 r2u 12(18)2(NOTE:5p 45p cm2 or 141 cm2 (rounded)1850 5 /18 rad.)(See Probs. 1.9 and 1.10.)1.7 Linear and Angular VelocityConsider an object traveling at a constant velocity along a circular arc of radius r. Let s be the length of thearc traveled in time t. Let 2 be the angle (in radian measure) corresponding to arc length s.Linear velocity measures how fast the object travels. The linear velocity, v, of an object is computed byarc lengthsn time t .Angular velocity measures how fast the angle changes. The angular velocity, (the lower-case Greekcentral angle in radiansuletter omega) of the object, is computed by v t.timeThe relationship between the linear velocity v and the angular velocity for an object with radius r isv rvwhere is measured in radians per unit of time and v is distance per unit of time.(NOTE:v and use the same unit of time and r and v use the same linear unit.)
6CHAPTER 1 Angles and ApplicationsEXAMPLE 1.7 A bicycle with 20-in wheels is traveling down a road at 15 mi/h. Find the angular velocity of the wheelin revolutions per minute.Because the radius is 10 in and the angular velocity is to be in revolutions per minute (r/min), change the linearvelocity 15 mi/h to units of in/min.15 mi # 5280 ft # 12 in # 1 hinmi 15,840h1 h1 mi 1 ft 60 minmin15,840 radradv 1584v r 10 minminv 15To change to r/min, we multiply by 1/2 revolution per radian (r/rad).v 1584792 rrad1584 rad # 1 r por 252 r/minmin1 min 2p radminEXAMPLE 1.8 A wheel that is drawn by a belt is making 1 revolution per second (r/s). If the wheel is 18 cm indiameter, what is the linear velocity of the belt in cm/s?1 2p radr 2p rad/s1s #1 1 rv rv 9(2p) 18p cm/s or 57 cm/s(See Probs. 1.11 to 1.15.)SOLVED PROBLEMSUse the directions for rounding stated on page 2.1.1 Express each of the following angles in radian measure:(a) 30 , (b) 135 , (c) 25 30 , (d) 42 24 35 , (e) 165.7 ,(f) 3.85 , (g) 205 , (h) 18 30 , (i) 0.21 (a) 30 30( /180) rad /6 rad or 0.5236 rad(b) 135 135( /180) rad 3 /4 rad or 2.3562 rad(c) 25 30 25.5 25.5( /180) rad 0.4451 rad(d) 42 24 35 42.41 42.41( /180) rad 0.7402 rad(e) 165.7 165.7( /180) rad 2.8920 rad(f) 3.85 3.85( /180) rad 0.0672 rad(g) 205 ( 205)( /180) rad 3.5779 rad(h) 18 30 18.01 ( 18.01)( /180) rad 0.3143 rad(i) 0.21 ( 0.21)( /180) rad 0.0037 rad1.2 Express each of the following angles in degree measure:(a) /3 rad, (b) 5 /9 rad, (c) 2/5 rad, (d) 4/3 rad, (e) /8 rad,(f) 2 rad, (g) 1.53 rad, (h) 3 /20 rad, (i) 7 rad(a) /3 rad ( /3)(180 / ) 60 (b) 5 /9 rad (5 /9)(180 / ) 100 (c) 2/5 rad (2/5)(180 / ) 72 / 22.92 or 22 55.2 or 22 55 12 (d) 4/3 rad (4/3)(180 / ) 240 / 76.39 or 76 23.4 or 76 23 24 (e) /8 rad ( /8)(180 / ) 22.5 or 22 30 (f) 2 rad (2)(180 / ) 114.59 or 114 35.4 or 114 35 24 (g) 1.53 rad (1.53)(180 / ) 87.66 or 87 39.6 or 87 39 36 (h) 3 /20 rad ( 3 /20)(180 / ) 27 (i) 7 rad ( 7 )(180 / ) 1260
7CHAPTER 1 Angles and Applications1.3 The minute hand of a clock is 12 cm long. How far does the tip of the hand move during 20 min?During 20 min the hand moves through an angle 120 2 /3 rad and the tip of the hand moves over adistance s r 12(2 /3) 8 cm 25.1 cm.1.4 A central angle of a circle of radius 30 cm intercepts an arc of 6 cm. Express the central angle inradians and in degrees.s61u r rad 11.46 3051.5 A railroad curve is to be laid out on a circle. What radius should be used if the track is to changedirection by 25 in a distance of 120 m?We are finding the radius of a circle on which a central angle 25 5 /36 rad intercepts an arc of120 m. Thenr 86412s p m 275 m u5p 361.6 A train is moving at the rate of 8 mi/h along a piece of circular track of radius 2500 ft. Through whatangle does it turn in 1 min?Since 8 mi/h 8(5280)/60 ft/min 704 ft/min, the train passes over an arc of length s 704 ft in1 min. Then s/r 704/2500 0.2816 rad or 16.13 .1.7 Assuming the earth to be a sphere of radius 3960 mi, find the distance of a point 36 N latitude fromthe equator.Since 36 /5 rad, s r 3960( /5) 2488 mi.1.8 Two cities 270 mi apart lie on the same meridian. Find their difference in latitude.s2703u r rad396044or3 54.4r1.9 A sector of a circle has a central angle of 50 and an area of 605 cm2. Find the radius of the circle.K 12r2u; therefore r 22K/u.r 2(605)2K4356 21386.56BuB (5p 18)B p 37.2 cm1.10 A sector of a circle has a central angle of 80 and a radius of 5 m. What is the area of the sector?11K 2r2u 2(5)2 a4p50p 2b m 17.5 m2991.11 A wheel is turning at the rate of 48 r/min. Express this angular speed in (a) r/s, (b) rad/min, and(c) rad/s.(a) 484rr48 r # 1 min s min1 min 60 s548 rr min1 min48 rr (c) 48min1 min(b) 48# 2p rad 96p rad or 301.6 rad1 rminmin# 1 min # 2p rad 8p rad or 5.03 rads60 s1 r5 s1.12 A wheel 4 ft in diameter is rotating at 80 r/min. Find the distance (in ft) traveled by a point on the rimin 1 s, that is, the linear velocity of the point (in ft/s).808p radr2p rad 80a b s min603 s
8CHAPTER 1 Angles and ApplicationsThen in 1 s the wheel turns through an angle 8 /3 rad and a point on the wheel will travel a distances r 2(8 /3) ft 16.8 ft. The linear velocity is 16.8 ft/s.1.13 Find the diameter of a pulley which is driven at 360 r/min by a belt moving at 40 ft/s.360rad2p radr 360a b s 12p smin60Then in 1 s the pulley turns through an angle 12 rad and a point on the rim travels adistance s 40 ft.s4020d 2r 2a b 2abft ft 2.12 ftu12p3p1.14 A point on the rim of a turbine wheel of diameter 10 ft moves with a linear speed of 45 ft/s. Find therate at which the wheel turns (angular speed) in rad/s and in r/s.In 1 s a point on the rim travels a distance s 45 ft. Then in 1 s the wheel turns through an angle s/r 45/5 9 rad and its angular speed is 9 rad/s.Since 1 r 2 rad or 1 rad 1/2 r, 9 rad/s 9(1/2 ) r/s 1.43 r/s.1.15 Determine the speed of the earth (in mi/s) in its course around the sun. Assume the earth’s orbit to bea circle of radius 93,000,000 mi and 1 year 365 days.In 365 days the earth travels a distance of 2 r 2(3.14)(93,000,000) mi.2(3.14)(93,000,000)In 1 s it will travel a distance s mi 18.5 mi. Its speed is 18.5 mi/s.365(24)(60)(60)SUPPLEMENTARY PROBLEMSUse the directions for rounding stated on page 2.1.16Express each of the following in radian measure:(a) 25 , (b) 160 , (c) 75 30 , (d) 112 40 , (e) 12 12 20 , (f) 18.34 Ans.1.17(a) 5 /36 or 0.4363 rad(b) 8 /9 or 2.7925 rad(c) 151 /360 or 1.3177 rad(d) 169 /270 or 1.9664 rad(e) 0.2130 rad(f) 0.3201 radExpress each of the following in degree measure:(a) /4 rad, (b) 7 /10 rad, (c) 5 /6 rad, (d) 1/4 rad, (e) 7/5 radAns.1.18On a circle of radius 24 in, find the length of arc subtended by a central angle of (a) 2/3 rad,(b) 3 /5 rad, (c) 75 , (d) 130 .Ans.1.19(a) 45 , (b) 126 , (c) 150 , (d) 14 19 12 or 14.32 , (e) 80 12 26 or 80.21 (a) 16 in, (b) 14.4 or 45.2 in, (c) 10 or 31.4 in,(d) 52 /3 or 54.4 inA circle has a radius of 30 in. How many radians are there in an angle at the center subtended by an arc of(a) 30 in, (b) 20 in, (c) 50 in?Ans.(a) 1 rad, (b)23rad, (c) 53 rad
9CHAPTER 1 Angles and Applications1.20Find the radius of the circle for which an arc 15 in long subtends an angle of (a) 1 rad, (b)(d) 20 , (e) 50 .Ans.1.211.24 rad or 71.05 or 71 3 40 or 125.7 cm2diameter 29.0 m3 or 9.4 m/s504 r/minIn grinding certain tools the linear velocity of the grinding surface should not exceed 6000 ft/s. Find the maximumnumber of revolutions per second of (a) a 12-in (diameter) emery wheel and (b) an 8-in wheel.Ans.1.31128 /3 or 134.04 mm2An automobile tire has a diameter of 30 in. How fast (r/min) does the wheel turn on the axle when the automobilemaintains a speed of 45 mi/h?Ans.1.306250 /9 or 2182 ftA flywheel of radius 10 cm is turning at the rate 900 r/min. How fast does a point on the rim travel in m/s?Ans.1.290.352 rad or 20 10 or 20.17 If the area of a sector of a circle is 248 m2 and the central angle is 135 , find the diameter of the circle.Ans.1.28rad or 7 9 36 or 7.16 Find the area of the sector determined by a central angle of 100 in a circle with radius 12 cm.Ans.1.2718Find the central angle necessary to form a sector of area 14.6 cm2 in a circle of radius 4.85 cm.Ans.1.26(e) 17.2 inFind the area of the sector determined by a central angle of /3 rad in a circle of diameter 32 mm.Ans.1.25(d) 43.0 in,A curve on a railroad track consists of two circular arcs that make an S shape. The central angle of one is 20 withradius 2500 ft and the central angle of the other is 25 with radius 3000 ft. Find the total length of thetwo arcs.Ans.1.24(c) 5 in,A train is traveling at the rate 12 mi/h on a curve of radius 3000 ft. Through what angle has it turned in 1 min?Ans.1.23(b) 22.5 in,rad, (c) 3 rad,The end of a 40-in pendulum describes an arc of 5 in. Through what angle does the pendulum swing?Ans.1.22(a) 15 in,23(a) 6000/ r/s or 1910 r/s, (b) 9000/ r/s or 2865 r/sIf an automobile wheel 78 cm in diameter rotates at 600 r/min, what is the speed of the car in km/h?Ans.88.2 km/h
CHAPTER 2Trigonometric Functionsof a General Angle2.1 Coordinates on a LineA directed line is a line on which one direction is taken as positive and the other as negative. The positivedirection is indicated by an arrowhead.A number scale is established on a directed line by choosing a point O (see Fig. 2.1) called the origin anda unit of measure OA 1. On this scale, B is 4 units to the right of O (that is, in the positive direction from O)and C is 2 units to the left of O (that is, in the negative direction from O). The directed distance OB 4 andthe directed distance OC 2. It is important to note that since the line is directed, OB BO and OC CO.The directed distance BO 4, being measured contrary to the indicated positive direction, and the directeddistance CO 2. Then CB CO OB 2 4 6 and BC BO OC 4 ( 2) 6.Fig. 2.12.2 Coordinates in a PlaneA rectangular coordinate system in a plane consists of two number scales (called axes), one horizontal andthe other vertical, whose point of intersection (origin) is the origin on each scale. It is customary to choosethe positive direction on each axis as indicated in the figure, that is, positive to the right on the horizontalaxis or x axis and positive upward on the vertical or y axis. For convenience, we will assume the same unitof measure on each axis.By means of such a system the position of any point P in the plane is given by its (directed) distances, calledcoordinates, from the axes. The x-coordinate of a point P (see Fig. 2.2) is the directed distance BP OA and theFig. 2.210
CHAPTER 2 Trigonometric Functions of a General Angle11Fig. 2.3y-coordinate is the directed distance AP OB. A point P with x-coordinate x and y-coordinate y will be denotedby P(x, y).The axes divide the plane into four parts, called quadrants, which are numbered (in a counterclockwisedirection) I, II, III, and IV. The numbered quadrants, together with the signs of the coordinates of a pointin each, are shown in Fig. 2.3.The undirected distance r of any point P(x, y) from the origin, called the distance of P or the radius vector of P, is given byr 2x2 y2Thus, with each point in the plane, we associate three numbers: x, y, and r.(See Probs. 2.1 to 2.3.)2.3 Angles in Standard PositionWith respect to a rectangular coordinate system, an angle is said to be in standard position when its vertexis at the origin and its initial side coincides with the positive x axis.An angle is said to be a first-quadrant angle or to be in the first quadrant if, when in standard position,its terminal side falls in that quadrant. Similar definitions hold for the other quadrants. For example, theangles 30 , 59 , and 330 are first-quadrant angles [see Fig. 2.4(a)]; 119 is a second-quadrant angle; 119 is a third-quadrant angle; and 10 and 710 are fourth-quadrant angles [see Fig. 2.4(b)].Fig. 2.4
12CHAPTER 2 Trigonometric Functions of a General AngleTwo angles which, when placed in standard position, have coincident terminal sides are called coterminalangles. For example, 30 and 330 , and 10 and 710 are pairs of coterminal angles. There is an unlimitednumber of angles coterminal with a given angle. Coterminal angles for any given angle can be found by addinginteger multiples of 360 to the degree measure of the given angle.(See Probs. 2.4 to 2.5.)The angles 0 , 90 , 180 , and 270 and all the angles coterminal with them are called quadrantal angles.2.4Trigonometric Functions of a General AngleLet u be an angle (not quadrantal) in standard position and let P(x, y) be any point, distinct from the origin, onthe terminal side of the angle. The six trigonometric functions of u are defined, in terms of the x-coordinate,y-coordinate, and r (the distance of P from the origin), as follows:sine u sin u yy-coordinate rdistancecotangent u cot u x-coordinatex yy-coordinatecosine u cos u x-coordinatex rdistancesecant u sec u rdistance xx-coordinatetangent u tan u y-coordinatey xx-coordinatecosecant u csc u rdistance yy-coordinateAs an immediate consequence of these definitions, we have the so-called reciprocal relations:sin 1/csc tan 1/cot sec 1/cos cos 1/sec cot 1/tan csc 1/sin Because of these reciprocal relationships, one function in each pair of reciprocal trigonometric functions hasbeen used more frequently than the other. The more frequently used trigonometric functions are sine, cosine,and tangent.It is evident from the diagrams in Fig. 2.5 that the values of the trigonometric functions of u change as uchanges. In Prob. 2.6 it is shown that the values of the functions of a given angle u are independent of thechoice of the point P on its terminal side.Fig. 2.5
13CHAPTER 2 Trigonometric Functions of a General Angle2.5 Quadrant Signs of the FunctionsSince r is always positive, the signs of the functions in the various quadrants depend on the signs of x and y.To determine these signs, one may visualize the angle in standard position or use some device as shown inFig. 2.6 in which only the functions having positive signs are listed.(See Prob. 2.7.)Fig. 2.6When an angle is given, its trigonometric functions are uniquely determined. When, however, the value1of one function of an angle is given, the angle is not uniquely determined. For example, if sin u 2, then 30 , 150 , 390 , 510 , . . . . In general, two possible positions of the terminal side are found; for example,the terminal sides of 30 and 150 in the above illustration. The exceptions to this rule occur when the angleis quadrantal.(See Probs. 2.8 to 2.16.)2.6Trigonometric Functions of Quadrantal AnglesFor a quadrantal angle, the terminal side coincides with one of the axes. A point P, distinct from the origin, onthe terminal side has either x 0 and y 0, or x 0 and y 0. In either case, two of the six functions will notbe defined. For example, the terminal side of the angle 0 coincides with the positive x axis and the y-coordinateof P is 0. Since the x-coordinate occurs in the denominator of the ratio defining the cotangent and cosecant,
CHAPTER 1 1 Angles and Applications 1.1 Introduction Trigonometry is the branch of mathematics concerned with the measurement of the parts, sides, and angles of a triangle. Plane trigonometry, which is the topic of this book, is restricted to triangles lying in a plane. Trigonometry is based on certain ratio
1 Right Triangle Trigonometry Trigonometry is the study of the relations between the sides and angles of triangles. The word “trigonometry” is derived from the Greek words trigono (τρ ιγων o), meaning “triangle”, and metro (µǫτρω ), meaning “measure”. Though the ancient Greeks, such as Hipparchus
1 Review Trigonometry – Solving for Sides Review Gr. 10 2 Review Trigonometry – Solving for Angles Review Gr. 10 3 Trigonometry in the Real World C2.1 4 Sine Law C2.2 5 Cosine Law C2.3 6 Choosing between Sine and Cosine Law C2.3 7 Real World Problems C2.4 8 More Real World Problems C2.4 9
Trigonometry Standard 12.0 Students use trigonometry to determine unknown sides or angles in right triangles. Trigonometry Standard 5.0 Students know the definitions of the tangent and cotangent functions and can graph them. Trigonometry Standard 6.0 Students know the definitions of the secant and cosecant functions and can graph them.
Key words: GeoGebra, concept teaching, trigonometry, periodicity. INTRODUCTION Trigonometry teaching Being one of the major subjects in high school mathematics curriculum, trigonometry forms the basis of many advanced mathematics courses. The knowledge of trigonometry can be also used in physics, architecture and engineering.
SWBAT: 1) Explore and use Trigonometric Ratios to find missing lengths of triangles, and 2) Use trigonometric ratios and inverse trigonometric relations to find missing angles. Day 1 Basic Trigonometry Review Warm Up: Review the basic Trig Rules below and complete the example below: Basic Trigonometry Rules: These formulas ONLY work in a right triangle.
TIPS4RM: Grade 10 Applied: Unit 2 – Trigonometry (August 2008) 2-1 Unit 2 Grade 10 Applied. Trigonometry . Lesson Outline . . Introduce the inverse trigonometric ratio key on a scientific calculator. Students check how close their answers are from the Day 2 Home Activity.
4 Trigonometry – An Overview of Important Topics So I hear you’re going to take a Calculus course? Good idea to brush up on your Trigonometry!! Trigonometry is a branch of mathematics that focuses on relationships between
aliments contenant un additif alimentaire des dispositions des alinéas a) et d) du paragraphe 4(1) ainsi que du paragraphe 6(1) de la Loi sur les aliments et drogues de même que, s'il y a lieu, des articles B.01.042, B.01.043 et B.16.007 du Règlement sur les aliments et drogues uniquement en ce qui a trait
