Algebra Chapers & : Trigonomeric Funcions of Angles and Real Numbers Chapers & : Trigonomeric Funcions of Angles and Real Numbers - Angle Measures Radians: - a uni (rad o measure he size of an angle. rad 80 o OR rad o 80 Divide boh Sides b 80 Convering Degree o Radian Using Graphing Calculaor. Sep : Se Mode o RADIAN. Sep : Ener Degree Sep : Specif Degree Uni and Conver MODE nd Selec Opion ANGLE APPS ENTER ENTER Convering Radian o Degree Using Graphing Calculaor. Sep : Se Mode o Degree. Sep : Ener Radian Sep : Specif Radian Uni and Conver MODE nd ANGLE Selec Opion APPS ENTER Eample : Conver he following ino radian. a. 90 o b. 0 o ENTER o rad 80 90 o 90 90 rad rad 80 80 o rad 80 0 o 0 0 rad rad 80 80 90 o rad. rad 0 o rad.09 rad Coprighed b Gabriel Tang B.Ed., B.Sc. Page 9.
Chapers & : Trigonomeric Funcions of Angles and Real Numbers Algebra c. o d. o o rad 80 o rad rad 80 80 o rad 80 o rad rad 80 80 o rad. rad o rad.9 rad e. 0 o f. 0 o o rad 80 0 o 0 0 rad rad 80 80 o rad 80 0 o 0 0 rad rad 80 80 0 o rad.9 rad 0 o rad. rad Eample : Conver he following ino degree. a. rad b. rad 80 o rad (80 o rad 0 o rad rad 80 o rad ( 80 o rad o c. rad rad 80 o rad ( 80 o rad 0 o d.. rad rad 80 o. rad degrees o 80.. 80. rad 9.9 o Page 0. Coprighed b Gabriel Tang B.Ed., B.Sc.
Algebra Chapers & : Trigonomeric Funcions of Angles and Real Numbers Sandard Posiion Angles: - angles ha can be defined on a coordinae grid. Iniial Arm: - he beginning ra of he angle, which is fied on he posiive -ais. Terminal Arm: - roaes abou he origin (0,0. - he sandard angle (θ is hen measured beween he iniial arm and erminal arm.. Posiive Angle: - angle formed b he erminal arm roaed couner-clockwise. 90 o Negaive Angle: - angle formed b he erminal arm roaed clockwise. 0 o erminal arm 80 o θ iniial arm 0o 0 o 80 o θ iniial arm 0o 0 o erminal arm 0 o 90 o Coerminal Angles: - angles form when he erminal arms ends in he same posiion. 90 o Quadrans: - he four pars of he Caresian Coordinae Grid. 90 o θ nd Quadran (, + s Quadran (+, + 80 o θ 0o 0 o 80 o rd Quadran (, 0o h Quadran (+, 0 o 0 o 0 o Coprighed b Gabriel Tang B.Ed., B.Sc. Page.
Chapers & : Trigonomeric Funcions of Angles and Real Numbers Algebra Eample : Given θ 0 o. Draw he angle θ in sandard posiion. Find and draw diagrams for wo oher angles which are coerminal o θ. θ 0 o 90 o 90 o θ 0 o 0 o + 0 o θ 80 o 80 o 0o 0 o 80 o 0o 0 o 0 o 0 o Reference Angle: - he acue angle beween he erminal arm and he -ais for an sandard posiion angle. Eample : Find he reference angle for he following angles in sandard posiion. a. b. c. θ 0 o 80 o 0 o θ ref 0 o o 80 o θ ref o θ o 0 o o θ ref o θ o - Assignmen: pg. #,,,,,,,, Page. Coprighed b Gabriel Tang B.Ed., B.Sc.
Algebra Chapers & : Trigonomeric Funcions of Angles and Real Numbers - The Uni Circle Uni Circle: - a circle wih a radius of and cenred a (0, 0 ha is drawn on a sandard Caresian grid. - he coordinaes of an poin of he uni circle can be found using is equaion, and he are relaed o some rigonomeric funcions such as cosine and sine (more in secion. Equaion of a Uni Circle 90 o + (0, 80 o (, 0 r θ (, (, 0 0 o, 0 o θ hea common variable for angles 0 o (0, Terminal Poin: - he coordinae of he uni circle of a paricular erminal arm s angle (. Reference Number: - also called he reference angle (. Eample : The poin P(, ½ is on he uni circle in he quadran I, find is -coordinae. + + (½ ± ± + ¼ In quadran I, > 0. r P(, ½ ¼ ½ ¾ Eample : The poin P (, is on he uni circle in he quadran IV, find is -coordinae. r P(, + ( + ( + ½ ½ ± ± ± ± In quadran IV, < 0. Coprighed b Gabriel Tang B.Ed., B.Sc. Page.
Chapers & : Trigonomeric Funcions of Angles and Real Numbers Algebra Eample : The poin P ( ⅔, is on he uni circle in he quadran III, find is -coordinae. ( ⅔ + r 9 + ± ± 9 P( ⅔, 9 In quadran III, < 0. 9 The Complee Uni Circle 0, 90 o (0,, 0 0,,,, 0 80 o (, 0 (, 0 0 o, 0 o 0, 0,,, 0 (0, 00 0 o,,, 0 Page. Coprighed b Gabriel Tang B.Ed., B.Sc.
Algebra Chapers & : Trigonomeric Funcions of Angles and Real Numbers Eample : Given he following erminal arm angle,, find he erminal poin P(, on he uni circle. a. c. P( ½, P(, ½ ½ r 0 ½ r (coerminal angles b. d. P(, ½ Eample : A erminal poin,,, is on an uni circle. Find he erminal poin of he following epression if i is on he same uni circle. ½ r (coerminal angles r (coerminal angles a. + b. c. P(, Q(, P(, + P(, P(, Eample : Find he reference number (reference angle given he following below. a. Q(, θ ref b. θ ref c. θ ref Q(, - Assignmen: pg. #,,,, 9,,, 9,,,,, ; Honour # Coprighed b Gabriel Tang B.Ed., B.Sc. Page.
Chapers & : Trigonomeric Funcions of Angles and Real Numbers Algebra - Trigonomeric Funcions of Real Numbers For an righ angle riangles, we can use he following simple rigonomeric raios or rigonomeric funcions. opposie sin θ hpoenuse adjacen cos θ hpoenuse opposie an θ adjacen Hpoenuse r θ Adjacen P (, Opposie SOH CAH TOA Wihin he uni circle, hese rig funcions (someimes called circular funcions are reduced o: sin θ r cos θ r an θ Reciprocal Trigonomeric Funcion: - he reciprocal of he regular rig funcions. Noe: For an θ, 0. - sine (sin urns ino cosecan (csc, Hence, an θ is - cosine (cos becomes secan (sec, undefined a 90 & 0 - and angen (an changed o coangen (co. Reciprocal Trigonomeric Funcions csc θ hpoenuse opposie hpoenus sec θ adjacen adjacen co θ opposie Wihin he uni circle, hese reciprocal rig funcions become csc θ r sec θ r co θ anθ ( Depending on he uni of he angle given (degree or radian, be sure ha our calculaor is se in DEGREE or RADIAN under he seings in our MODE menu! Noe: For co θ, 0 and 0. Hence, co θ is undefined a 0, 90, 80, 0 and 0. The coordinaes (, are he same as (cos θ, sin θ of an angle θ in he uni circle. nd Quadran s Quadran Sin All sin θ + cos θ an θ sin θ + cos θ + an θ + S A Tan Cos sin θ cos θ an θ + sin θ cos θ + an θ T C rd Quadran h Quadran Page. Coprighed b Gabriel Tang B.Ed., B.Sc.
Algebra Chapers & : Trigonomeric Funcions of Angles and Real Numbers Eample : Using he uni circle, find he eac value of he rigonomeric funcion a he given real number angle. a. sin 0 b. cos c. an d. csc 0 e. co A 0, P(, ½ sin 0 sin 0 ½ f. sin 00 g. an A 00, P(½, sin 00 h. cos i. sec A j. co A, P(0, co( P(, an co ( ( Eample : From he informaion given below, deermine which quadran he erminal poin has o be a. a. sin < 0 and cos < 0 sin < 0 means < 0 ; cos < 0 means < 0 Since boh and is negaive in he quadran III, b. co < 0 and sec > 0 co < 0 means an < 0 (Quadrans II and IV ; A, P(, cos( cos( A, P(, A, P(½, an( an( ( Coprighed b Gabriel Tang B.Ed., B.Sc. Page. ( A 0, P( ½, csc( ( csc( A, P(, ½ an( ( an( an( sin 00 ( ( ( 0 sec( an( cos ( ( 0 co( sec( undefined Eample : Given he erminal poin, P,, find he values of he rigonomeric funcions. > 0 and < 0 means P is a quadran IV csc csc sin sin ( cos cos sec r mus be in quadran III. sec > 0 means cos > 0 (Quadrans I and IV mus be in quadran IV. ( an ( sin θ cos θ an θ + rd Quadran A, P(0, cos ( sin θ cos θ + an θ P(, ½, sec ( co ( h Quadran (
Chapers & : Trigonomeric Funcions of Angles and Real Numbers Algebra Eample : Find he values of he rigonomeric funcions of from he given informaion. a. sin and an < 0 sin > 0 means > 0 (Quadrans I and II ; an < 0 (Quadrans II and IV is in quadran II. P(, r + + ( + ± ± In quadran II, < 0. nd Quadran sin θ + cos θ an θ sin sin cos cos an ( ( an csc ( sec ( co ( ( csc sec co b. cos and sin < 0 cos < 0 means < 0 (Quadrans II and III ; sin < 0 means < 0 (Quadrans III and IV is in quadran III. sin θ cos θ an θ + rd Quadran P(, + + ( sin sin 9 + 9 9 cos cos an ( ( r an ± ± 9 In quadran III, < 0. csc ( sec ( co ( ( csc sec co - Assignmen: pg. #, 9,,,,,,,,,, 8; Honour #9,,, 8 Page 8. Coprighed b Gabriel Tang B.Ed., B.Sc.
Algebra Chapers & : Trigonomeric Funcions of Angles and Real Numbers - Trigonomeric Graphs a sin k( + b + c a cos k( + b + c a Ampliude c Verical Displacemen (how far awa from he ais b Horizonal Displacemen (Phase Shif b > 0 (shifed lef b < 0 (shifed righ k number of complee ccles in Period Range Minimum Maimum o 0 k k sin cos Eamples: sin ( + cos ( a k ccles in b righ a k ccles in b c Period Range: d Period Range: Phase Shif Period Phase Shif a c Period k complee ccles in a c k complee ccles in Coprighed b Gabriel Tang B.Ed., B.Sc. Page 9.
Chapers & : Trigonomeric Funcions of Angles and Real Numbers Algebra Graphing Trigonomeric Funcions. Idenif he ampliude, phase shif, number of complee ccles in, and verical displacemen.. Calculae he period and he range.. From he period and phase shif, deermine he inerval needed along he -ais.. From he verical displacemen and he range, deermine he inerval needed on he -ais.. Divide each period ino four secions, use some fi poins from he original sine and cosine graph (such as 0,,,, and along wih he range o plo some poins. Then connec he dos. Eample : Find he ampliude, period, phase shif, and verical displacemen of he funcion and skech is graph over a leas one period. amp ( a. sin + amp c up b. sin ( + c 0 (no ver. disp. Range: Range: b 0 (no phase shif k ccle in b o he lef k ccle in Period k Period Period k Period c amp Period Sine sars a midline and move up. b (lef amp Period Sine sars a midline and move down because a < 0 (graph had refleced off -ais. c. ½ cos ( ampliude ½ c 0 (no ver. disp. d. cos ( + Range: ½ ½ cos ( + b (righ Period k k ccles in Period b (lef b (lef Period ampliude c ( down Range: k ccles in Period k Period Period b (righ amp ½ Cosine sars a maimum and move down. Cosine sars a maimum and move down. c amp Page 0. Coprighed b Gabriel Tang B.Ed., B.Sc.
Algebra Chapers & : Trigonomeric Funcions of Angles and Real Numbers Eample : The graph of one complee period of a sine or cosine curve is given. Find he ampliude, period, phase shif, and verical displacemen. Wrie an equaion ha represen he curve in he form of sine and cosine funcions. a. b. Period Cosine sars a maimum and move down. b (righ amp Sine sars a he midline and move down. Sine sars a he midline and move down. b (righ b (lef Cosine sars a maimum and move down. Period Period amp. c Period Period k k k amp. a c 0 (no ver. disp. For cosine, b 0. cos ( For sine, b. sin ( To Graph Trig Funcions on he Graphing Calculaor:. Se calculaor o Radian in he MODE screen.. Ener he equaion in he Y screen.. Press ZOOM and selec ZTrig (opion. Period Period k k k amp.. a. For cosine, b. c. cos [( ] For sine, b.. sin [( ] Eample : Graph ¼, ¼ and ¼ cos. How are he graphs relaed? In Radian Mode, ener equaions in Y ZOOM Selec ZTrig The Trig equaion (Y is bounded b he quadraic equaions (Y & Y. This is because he Trig equaion consiss of boh a quadraic funcion (¼ and a rig funcion (cos (. Eample : Find he range of cos + cos. Ener equaion in Y ZOOM Selec ZTrig From he graph, he Range is Eample : Find all soluions of sin 0. for [0, ]. Ener equaion in Y Run Inersec wice in nd TRACE - Assignmen: pg. 9 #,,, 9,,,,,,,, 9,, ; Honour #9, 8 Coprighed b Gabriel Tang B.Ed., B.Sc. Page.
Chapers & : Trigonomeric Funcions of Angles and Real Numbers Algebra - Modeling Harmonic Moion Someimes, a descripion of he periodic paern (harmonic moion if he paern applies o a moving objec is given. In such case, i is ver imporan o deermine he feaures of he graph (ampliude, period, horizonal displacemen, and verical displacemen. The will be used o generae he parameers needed for he basic rigonomeric funcion, a sin [ω ( + b] + c, or a cos [ω ( + b] + c. (Noe: k he number of complee ccles in is now replaced b ω. Period: - he amoun of ime needed o complee one ccle. Frequenc: - he number of ccles per uni of ime. (The longer is he period; he smaller he frequenc. a sin [ω ( + b] + c a cos [ω ( + b] + c a Ampliude c Verical Displacemen (disance beween mid-line and -ais b Horizonal Displacemen (Phase Shif b > 0 (shifed lef b < 0 (shifed righ ω number of complee ccles in Period ω Range Minimum Maimum Frequenc Eample : A mechanical pendulum has a heigh of m off he ground. When i swings o he highes poin, is heigh is m off he ground. I makes complee swings per minue, and he saring poin is on he righ side of he res posiion. a. Wha is he period of he pendulum? b. Draw a graph o describe he heigh of he pendulum versus ime for complee ccles. c. Eplain all he feaures of he graph and deermine he equaion of heigh in erms of ime. d. Find he heigh of he pendulum a 0. seconds. e. A wha ime(s will he heigh of he pendulum be a. m during he firs complee ccle? ω a. Frequenc swings / min swings 0 seconds Frequenc ¼ swing/ sec Period ime in seconds ccle(or swing seconds swing Period sec / ccle b. Heigh of Pendulum Swings versus Time Heigh (m 8 0 0 8 9 0 Time (seconds Page. Coprighed b Gabriel Tang B.Ed., B.Sc.
Algebra Chapers & : Trigonomeric Funcions of Angles and Real Numbers c. Characerisics of he Graph Ampliude a m (how far he heigh is varied from one side of he swing o he res posiion Verical Displacemen c m (he average heigh of he pendulum Range: m h m (he min and ma heighs of he pendulum rad Period sec (ime o complee one full swing Period ω ω Period For cosine funcion, Horizonal Translaion b 0 For sine funcion, Horizonal Translaion b second (righ a ω b c ω 0 (for cos (for sin h cos ( + h sin [ ( ] + d. Heigh a 0. seconds. Ener equaion in Radian Mode. Run TRACE. Window Seings: : [0,, ] and : [ 0, 8, ] e. When will he pendulum reach. m during he firs complee ccle?. Ener Y equaion as.. Run Inersec wice on he firs ccle. Heigh.8 m 0.89 seconds and. seconds Eample : The London Ee is one of he larges ferris wheels. I has a diameer of m and he boom of he wheel passes m above ground. A complee revoluion akes 0 minues and he visiors are reaed wih an uninerruped view of he ci as far ou as 0 km ( miles. Deermine he equaion of he visior s heigh as a funcion of ime saring a he lowes poin of he wheel. v Ampliude Half he diameer m a. m Ver. Disp. Heigh beween ground & mid-line c. m + m Period 0 min ω ω 0 ω For cosine funcion, mins o ge o highes poin b For sine funcion,. mins o be half wa up b. h. cos [ ( ] + 8. h. sin [ (.] + 8. - Assignmen: pg. 9 #,, 9,,, 0, ; Honour # Coprighed b Gabriel Tang B.Ed., B.Sc. Page.