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# COLLEGE ALGEBRA (QL)(SSS) MATH 1050

Utah State University

GPA 3.66

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This 28 page Class Notes was uploaded by Benton Yundt on Wednesday October 28, 2015. The Class Notes belongs to MATH 1050 at Utah State University taught by Rong Xia in Fall. Since its upload, it has received 7 views. For similar materials see /class/230418/math-1050-utah-state-university in Mathematics (M) at Utah State University.

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Date Created: 10/28/15

Refresher Course Math 1050 and 1060 Notes Set 8 ltlt An es and Their Measure gtgt The startingposition ofthe ra o an position a ei rotation is called the terminal side The endpoint ofthe ray is the vertex Defn An angle is determined by rotating a ray halfrllne about its endpoint yisthe sie ftheanle dthe Note Positive angles are generated by counterclockwise rotation and negative angles by clockwise rotation Note Just as there are multiple measurement systems and units of measure for m m 7m t v nhm t v kil m t v t t mi quotpinches degrees and 2 rd mil l measuring angles Two commonly used units of angle measure ar radians Not Ni V Fmili rm However we must adapt to using radians because many important results in calculus M l villi l l assume angle measurements are given in radians Ni wh th di m the corresponding arc on the unit circle L5 a circle ofmdlus i That is is an angle that unlt The gure shows an ang e whose measure is 1 radian Angles are closely related to rotations Ln the figure 1 l 1 m l Oslthel mates count r clockwise rotation ofthep int 1 0 since the clicumf rence of the unit ircle is 27 one halfway around the unit circle 1 a rotation of n r diam A quarteirmtatlon an eighthrmtatlont and a twel hrmtatlon meaxur 7 and Texpectlvelyiiecallthat when unit are omitted radian are axxumed Page 1 of14 Refresher Course Math 1050 and 1060 Notes Set 8 Note See httpwwwwucronlmecomobjectxmdextjaxpwbledmhlSU1 N0 the conexpondmg angwmmnon THE UNIT CIRCLE n N Smce 27 mdxan conexpond to one complete xevoluuon degree andmdxan are related by the equauon 360 27 md and 180 md SO 1 l md and 1md 180 7 Note Conversions Between Degrees and Radian 7r rad 180 1 To conven degree to mdxam muluply degree by 180 2 To conven mam to degree muluplymdxan by 7 rad PageZofM Refresher Course Math 1050 and 1060 Notes Set 8 Example Convert 270 to radians Example Convert 2 radians to degrees Defn Two angles that have the same initial and terminal sides are called coterminal Note You can find an angle that is coterminal to a given angle 6 by adding or subtracting 360 27 one revolution Example Find two coterminal angles one positive and one negative for 6 120 Defn Two positive angles are complementary complements of each other if their sum is 90 5 2 Page 3 of 14 Refresher Course Math 1050 and 1060 Notes Set 8 Defn Two positive angles are supplementary supplements of each other if their sum is 180 7 Example Find the complement and supplement of 148 ltlt Right Traingle Trigonometry gtgt Pythagorean Theorem 0 A I Note Let 6 be an acute angle of a right triangle see figure Six functions sine ine tangent cosecanti ant cotangent called trigonometric functions are defined as follows sin cos tan hyp csc 6 sec 6 cot 6 Opp Example Find the values of sin 45 cos 45 and tan 45 Page 4 of 14 Refresher Course Math 1050 and 1060 Notes Set 8 Example Find the values of sin 60 cos 60 sin 30 and cos 30 Example Use the results from the previous two examples to fill the blanks in the table COS tan Example Find tan 6 if cos 0 Page 5 of 14 Refresher Course Math 1050 and 1060 Notes Set 8 Note From the main definitions of the six trigonometric functions we see that csc0 sec0 cot0 sin 0 tan 0 These relationships are called identities since they hold for all values of 6 for which both sides of the equations are defined There are many other trigonometric identities For example cos 0 cos0 did hyp adj adj cos0 hyp cot 0 L f tan 0 Sin 0 Another fundamental identity called the Pythagorean Identity is derived from the P ha orean Theorem Yt g UPPY ddj2 WY 011112 adj2 hylv2 3 2 2 MP hyp UPPY WV 1 hylv2 hylv2 2 2 1 hyp hyp 3 sin20cos20l 3 3 Note sin2 0 is an abbreviated notation for sin 0f similar notation applies to the other trigonometric functions Note Page 6 of 14 Refresher Course Math 1050 and 1060 Notes Set 8 Example Find tanH if cos0 using identities Example Use trigonometric identities to transform one side of the equation into the 2 sec 0 tan 0sec 0 tan 0 1 other 0lt0lt Page 7 of 14 Refresher Course Math 1050 and 1060 Notes Set 8 ltlt Tri onometric Functions ofAn An e gtgt Note m 4 page are in each taxe became 51 an acute angle of a right mangle We definitiomiand comequently With the identitie we have obxelved Notice if is an acute an e cos n and sin n 7 correspond to the x and ycoorilinates respectively of the point on the unit circle that corresponds to n co For example Th1 obxewation motivate the folloWing deflnltlom co 9 ecoordinate ofthe point on the unit circle conexpondmg to 9 m9 ycooidimte of the point on the unit circle conexpondmg to 9 the mat of all real numbelx 2 2 Noticethe Pythagorean dentltyholdx for all 9 X W i S 5 9 1 equation for the unit circle previou ection we define the other four trigonometric hinctiom a follow Pythagorean Identity PageSofM Refresher Course Math 1050 and 1060 Notes Set 8 Note Considering the nature of the circular definitions of the trigonometric functions one would be wise to become extremely familiar with the unit circleito the point that one can readily reconstruct it or desired portions of it on paper andor mentally Unit Circle Page 9 of 14 Refresher Course Math 1050 and 1060 Notes Set 8 Note Let 6 be an angle in standard position with x y a point on the terminal side of 6 and rx2y2 0 Then sin0l cos0i r r tan0l x 0 cot0 y 0 x y sec0 x 0 csc0 y 0 x y Example Let 34 be a point on the terminal side of 6 Find the sine cosine and tangent of 6 Example Given that tan0 75 and cosH gt 0 Find sing and sec l Defn Let 6 be an angle in standard position Its reference angle is the acute angle 0 39 formed by the terminal side of 6 and the horizontal axis Page 10 of 14 Refresher Course Math 1050 and 1060 Notes Set 8 Example Find the reference angle 0 39 a 6 300 b 6 23 c a 135 Evaluating Trigonometric Functions of Any Angle To find the value of a trigonometric function of any angle 6 1 Determine the function value for the associated reference angle 0 39 2 Depending on the quadrant in which 6 lies affix the appropriate sign to the function value Example Evaluate 47r a cos 7 3 b tan 210 Page 11 of 14 Refresher Course Math 1050 and 1060 Notes Set 8 Example Evaluate cscllT Example Find all solutions to the given equation within the specified interval sin 3x 7r 7r Page 12 of 14 Refresher Course Math 1050 and 1060 Notes Set 8 ltlt Graphs of Sine and Cosine Functions gtgt Example Sketch the graph of y sin x by plotting points Example Sketch the graph of y cos x by plotting points Example Sketch the graphs of y 3 sin x and y sin x Page 13 of 14 Refresher Course Math 1050 and 1060 Notes Set 8 DLfn A function f is periodic if there exists a positive real number 0 such that f t C f I for all t in the domain of f The least number 0 for which f is periodic is called the period of f Note Let b be a positive real number The period of y a sin bx and y a cos bx is given by Period 2 27 b Example Sketch the graph of y sin Note The graphs of y a sinbx c and y a cosbx c have the following characteristics Assume b gt 0 Amplitudezlal Period 2 27 The left and right endpoints of a one cycle interval can be determined by solving the equations bx c 0 and bx c 27 Example Sketch the graph of y isin x Page 14 of 14 Refresher Course Math 1050 and 1060 Nola Set 8 Fall 2007 ltlt An es and Their Measure gtgt Defn An angle is determined by rotating aray halfrllne about its endpoint The startingposition ofthe ray is the si e of the an 1e and the position a ei rotation is called the terminal side The endpoint ofthe ray is the vertex Note Positive angles are generated by counterclockwise rotation and negative angles by clockwise rotation Note Just as there are multiple measurement systems and units of measure for m tn 7m t v nhm t v kil m t v t t nd Milnches rd mil t measuring angles Two commonly used units of angle measure ar degrees and 2 radians Not W t Fmili viti However we must adapt to using radians because many important results in calculus M l quotin l i assume angle measurements are given in radians Ni wh th dt n7 the corresponding arc on the unit circle tie a circle ofmdlus i That is is an angle that unit The gure shows an ang e whose measure is 1 radian Angles are closely related to rotations Ln the figure halfway around the unit circle I a rotation of a r diam A quarteirmtatlon an eighthrmtatlont and a tweltthrotation meaxur 7 and iexpectlvelyiiecallthat when unit are omitted radian are axxumed Page 1 of14 Refresher Course Math 1050 and 1060 Nola Set 8 Fall 2007 Note See httpwwwwucronlmecomobjectxmdextjaxpwbledmhlSU1 N0 the conexpondmg angwmmnon THE UNIT CIRCLE n N Smce 27 mdxan conexpond to one complete xevoluuon degree andmdxan are related by the equauon 360 27 md and 180 md SO 1 l md and 1md 180 7 Note Conversions Between Degrees and Radian 7r rad 180 1 To conven degree to mdxam muluply degree by 180 2 To conven mam to degree muluplymdxan by 7 rad PageZofM Refresher Course Math 1050 and 1060 Notes Set 8 Fall 2007 Example Convert 270 to radians Example Convert 2 radians to degrees Defn Two angles that have the same initial and terminal sides are called coterminal Note You can find an angle that is coterminal to a given angle 6 by adding or subtracting 360 27 one revolution Example Find two coterminal angles one positive and one negative for 6 120 Defn Two positive angles are complementary complements of each other if their sum is 90 5 2 Page 3 of 14 Refresher Course Math 1050 and 1060 Notes Set 8 Fall 2007 Defn Two positive angles are supplementary supplements of each other if their sum is 180 7 Example Find the complement and supplement of 148 ltlt Right Traingle Trigonometry gtgt Pythagorean Theorem 0 A I Note Let 6 be an acute angle of a right triangle see figure Six functions sine ine tangent cosecanti ant cotangent called trigonometric functions are defined as follows sin cos tang hyp csc 6 sec 6 cot 6 Opp Example Find the values of sin 45 cos 45 and tan 45 Page 4 of 14 Refresher Course Math 1050 and 1060 Notes Set 8 Fall 2007 Example Find the values of sin 60 cos 60 sin 30 and cos 30 Example Use the results from the previous two examples to fill the blanks in the table COS tan Example Find tan 6 if cos 0 Page 5 of 14 Refresher Course Math 1050 and 1060 Notes Set 8 Fall 2007 Note From the main definitions of the six trigonometric functions we see that csc0 sec0 cot0 sin 0 tan 0 These relationships are called identities since they hold for all values of 6 for which both sides of the equations are defined There are many other trigonometric identities For example cos 0 cos0 did hyp adj adj cos0 hyp cot 0 L f tan 0 Sin 0 Another fundamental identity called the Pythagorean Identity is derived from the P ha orean Theorem Yt g UPPY ddj2 WY 011112 adj2 hyiv2 3 2 2 MP hyp UPPY WV 1 hurl2 hyiv2 2 2 1 hyp hyp 3 sin20cos201 3 3 Note sin2 0 is an abbreviated notation for sin 0f similar notation applies to the other trigonometric functions Note Page 6 of 14 Refresher Course Math 1050 and 1060 Notes Set 8 Fall 2007 Example Find tanH if cos0 using identities Example Use trigonometric identities to transform one side of the equation into the 2 sec 0 tan 0sec 0 tan 0 1 other 0lt0lt Page 7 of 14 Refresher Course Math 1050 and 1060 Nola Set 8 Fall 2007 ltlt Tri onometric Functions ofAn An e gtgt Note m 4 page are in each taxe became 51 an acute angle of a right mangle We definitlomiand comequently With the identltle we have obxelved Notice if is an acute an e cos n and sin n 7 correspond to the x and ycoorilinates respectively of the point on the unit circle that corresponds to n co For example Th1 obxewation motivate the folloWing deflnltlom co 9 ecoordinate ofthe point on the unit circle conexpondmg to 9 m9 ycooldimte of the point on the unit circle conexpondmg to 9 the mat of all real numbelx 2 a Noticethe Pythagorean dentltyholdx for all 9 X W i S 5 9 1 equation for the unit circle previou ection we define the other foul trigonometric hinctiom a follow Pythagorean Identity PageSofM Refresher Course Math 1050 and 1060 Notes Set 8 Fall 2007 Note Considering the nature of the circular definitions of the trigonometric functions one would be wise to become extremely familiar with the unit circleito the point that one can readily reconstruct it or desired portions of it on paper andor mentally Unit Circle Page 9 of 14 Refresher Course Math 1050 and 1060 Notes Set 8 Fall 2007 Note Let 6 be an angle in standard position with x y a point on the terminal side of 6 and rx2y2 0 Then sin0l cos0i r r tan0l x 0 cot0 y 0 x y sec0 x 0 csc0 y 0 x y Example Let 34 be a point on the terminal side of 6 Find the sine cosine and tangent of 6 Example Given that tan0 75 and cosH gt 0 Find sing and secH Defn Let 6 be an angle in standard position Its reference angle is the acute angle 0 39 formed by the terminal side of 6 and the horizontal axis Page 10 of 14 Refresher Course Math 1050 and 1060 Notes Set 8 Fall 2007 Example Find the reference angle 0 39 a 6 300 b 6 23 c a 135 Evaluating Trigonometric Functions of Any Angle To find the value of a trigonometric function of any angle 6 1 Determine the function value for the associated reference angle 0 39 2 Depending on the quadrant in which 6 lies affix the appropriate sign to the function value Example Evaluate 47r a cos 7 3 b tan 210 Page 11 of 14 Refresher Course Math 1050 and 1060 Notes Set 8 Fall 2007 Example Evaluate cscllT Example Find all solutions to the given equation within the specified interval sin 3x 7r 7r Page 12 of 14 Refresher Course Math 1050 and 1060 Notes Set 8 Fall 2007 ltlt Graphs of Sine and Cosine Functions gtgt Example Sketch the graph of y sin x by plotting points Example Sketch the graph of y cos x by plotting points Example Sketch the graphs of y 3 sin x and y sin x Page 13 of 14 Refresher Course Math 1050 and 1060 Notes Set 8 Fall 2007 DLfn A function f is periodic if there exists a positive real number 0 such that f t C f I for all t in the domain of f The least number 0 for which f is periodic is called the period of f Note Let b be a positive real number The period of y a sin bx and y a cos bx is given by Period 2 27 b Example Sketch the graph of y sin Note The graphs of y a sinbx c and y a cosbx c have the following characteristics Assume b gt 0 Amplitudezlal Period 2 27 The left and right endpoints of a one cycle interval can be determined by solving the equations bx c 0 and bx c 27 Example Sketch the graph of y isin x Page 14 of 14

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