Graphs of Sine and Cosine Functions

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1 Section.5 Graphs o Sine and Cosine Functions ln Domain: All real numers ecept 0 -intercepts: ±, 0 Vertical asmptote: 0 9 (, 0) (, 0) lo 0 To ind the -intercept, let 0: 0 lo 0 0 -intercepts:, 0 To ind the -intercept, let 0: lo 0 lo intercepts: 0, 0.0 The vertical asmptote is the horizontal asmptote o lo 0 translated two units to the let. Vertical asmptote: Domain: All real numers such that > (, 0) Section.5 Graphs o Sine and Cosine Functions You should e ale to raph a sin c and a cos c. Assume > 0. a Shit: Solve c 0 and c. Ke increments: (period) Vocaular Check. ccle. amplitude.. phase shit 5. vertical shit

2 7 Chapter Trionometr. sin. cos. 5 cos a 5 5. sin a 5. sin. cos a 7. sin 8. cos a 9. sin sin 8. 8 cos. 5 cos 8 a a 5. sin. cos a 5. sin sin The raph o is a horizontal shit to the riht units o the raph o a phase shit.. cos, cos 7. cos 8. sin, sin is a horizontal shit o units cos is a relection o aout the to the let. -ais. The raph o is a relection in the -ais o the raph o. 9. cos 0. cos The period o is twice that o. sin, sin. The period o is one-third the period o. sin sin The raph o is a vertical shit three units upward o the raph o.

3 Section.5 Graphs o Sine and Cosine Functions 75. cos, cos is a vertical shit o two units downward.. The raph o has twice the amplitude as the raph o. The period is the same.. The period o is one-third the period o. 5. The raph o is a horizontal shit units to the riht o the raph o.. Shit the raph o two units upward to otain the raph o. 7. sin Smmetr: oriin 5 Ke points: Intercept Minimum Intercept Maimum Intercept 0, 0,, 0, 0, 0 5 Since sin, enerate ke points or the raph o multiplin the -coordinate o each ke point o. 8. sin Smmetr: oriin Ke points: Intercept Maimum Intercept Minimum Intercept 0, 0,, 0,, 0 Since sin the raph o is the raph o, ut stretched horizontall a actor o., Generate ke points or the raph o multiplin the -coordinate o each ke point o. 9. cos Smmetr: -ais Ke points: Maimum Intercept Minimum Intercept Maimum 0,, 0,, 0, Since cos, the raph o is the raph o, ut translated upward one unit. Generate ke points or the raph o addin to the -coordinate o each ke point o.

4 7 Chapter Trionometr 0. cos Smmetr: -ais Ke points: Maimum Intercept Minimum Intercept Maimum 0,, 0,, 0, Since cos, the raph o is the raph o, ut i) shrunk horizontall a actor o, ii) shrunk verticall a actor o, and iii) relected aout the -ais. Generate ke points or the raph o i) dividin the -coordinate o each ke point o, and ii) dividin the -coordinate o each ke point o.. sin Smmetr: oriin Ke points: Intercept Minimum Intercept Maimum Intercept 5 0, 0,, 0,, 0 Since the raph o is the raph o, ut translated upward three units. sin, Generate ke points or the raph o addin to the -coordinate o each ke point o.. sin Smmetr: oriin Ke points: Intercept Maimum Intercept Minimum Intercept 0, 0,, 0, Since sin, the raph o is the raph o, ut translated downward three units. Generate ke points or the raph o sutractin rom the -coordinate o each ke point o., 0 8

5 Section.5 Graphs o Sine and Cosine Functions 77. cos Smmetr: -ais Ke points: Maimum Intercept Minimum Intercept Maimum 0,, 0,, 0, Since cos, the raph o is the raph o, ut with a phase shit (horizontal translation) o. Generate ke points or the raph o shitin each ke point o units to the let.. cos Smmetr: -ais Ke points: Minimum Intercept Maimum Intercept Minimum 0,, 0,, 0, Since cos, the raph o is the raph o, ut with a phase shit (horizontal translation) o. Generate ke points or the raph o shitin each ke point o units to the riht. 5. sin. sin Ke points: 0, 0,,,, 0,,,, 0 Ke points: 0, 0,,,, 0,,,, 0 7. cos 8. cos Ke points: 0,,, 0,,,, 0,, Ke points: 0,,, 0,,,, 0,,

6 78 Chapter Trionometr 9. cos 0. sin Ke points: 0,,, 0,,,, 0,, Ke points: 0, 0, 8,, 8,,, 0, 0,. cos. Ke points: 0,,, 0,,,, 0 sin 8 Ke points: 0, 0,,,, 0,,, 8, 0. sin ; a,, c 0 Ke points: 0, 0,,,, 0, 9,,, 0. 0 cos 0 Ke points: 0, 0,, 0,, 0, 9, 0,, sin ; a,, c Shit: Set 0 and 9 Ke points:, 0,,, 5, 0, 7,, 9, 0

7 Section.5 Graphs o Sine and Cosine Functions 79. sin 7. cos Shit: Set 0 and Shit: Set 0 and Ke points:, 0,,,, 0, 5,,, 0 Ke points:,,, 0, 0,,, 0,, 8. cos 9. Shit: Set 0 and Ke points:,,, 0,,, 7 5, 0, 7, sin Ke points: 5 0,,,,,, 9,,, 50. t 5 cos 5 Ke points: 0,,,,, 8, 8,,, t

8 80 Chapter Trionometr 5. cos Vertical shit two units upward Ke points: 0 0,., 0,, 0,.9, 0,, 0,. cos Ke points: 0,,,,, 5,,,, cos Shit: Set 0 and Ke points:, 0,,, 0,,,,, Shit: Set 0 and Ke points: cos 7, 8,,,, 0, 5,, 7, cos ; a,, c Shit: Ke points: 0 and 9,,, 0, 5,, 7, 0, 9,

9 Section.5 Graphs o Sine and Cosine Functions 8 5. cos Shit: Set 0 and Ke points:,,, 0, 0,,, 0,, 57. sin 58. sin cos 8 0. cos. 0. sin sin 0 t a cos d. a Vertical shit one unit upward o cos d Thus, cos. a cos d cos 0 d d a, d 5. a cos d. 8 0 Since is the raph o cos relected in the -ais and shited verticall our units upward, we have a and d. Thus, cos. a cos d Relected in the -ais: a cos 0 d d a, d

10 8 Chapter Trionometr 7. a sin c 8. a sin c Since the raph is relected in the -ais, we have a. a Phase shit: c 0 Thus, sin. a Phase shit: c 0 a,, c 0 9. a sin c 70. a Phase shit: c 0 when Thus, sin c 0 c. a sin c Phase shit: a c a,, c c 7. sin 7. cos In the interval,, sin when 5,, 7,. when, 7. t 0.85 sin v.00 (a) Time or one ccle sec () Ccles per min 0 t ccles per min 0.5 (c) 0.85; Ke points: 0, 0,, 0.85,, 0, 9, 0.85,, t v.75 sin (a) () Period seconds ccle 0 seconds 5 ccles per minute seconds minute (c) v 5 7 t

11 Section.5 Graphs o Sine and Cosine Functions sin 880t7. (a) seconds () 0 ccles per second p P 00 0 cos 5t (a) () 5 5 seconds hearteat 0 seconds 50 hearteats per minute 5 seconds minute 77. (a) a hih low (c) 00 p hih time low time 7 () p c 7 c 7 The model is a ood it..7 d hih low Ct cos t The model is a ood it. (d) Tallahassee averae maimum: Chicao averae maimum: 5.55 The constant term, d, ives the averae maimum temperature. (e) The period or oth models is months. This is as we epected since one ull period is one ear. () Chicao has the reater variailit in temperature throuhout the ear. The amplitude, a, determines this variailit since it is hih temp low temp. 78. (a) and (c) Percent o moon s ace illuminated Reasonal ood it (d) Period is 9. das. (e) March Da o the ear % The Naval oservator sas that 50% o the moon s ace will e illuminated on March, 007. () Vertical shit: d a Horizontal shit: C 0 7. (averae lenth o interval in data) C 0.9 sin0. 0.9

12 8 Chapter Trionometr 79. C 0.. sin t (a) Period (a) () (c) Period 5 Yes, this is what is epected ecause there are 5 das in a ear. The averae dail uel consumption is iven the amount o the vertical shit (rom 0) which is iven the constant The wheel takes minutes to revolve once. () 50 eet (c) The radius o the wheel is 50 eet. 0 minutes The consumption eceeds 0 allons per da when < < False. The raph o sin( )8. is the raph o sin() False. cos has an amplitude that is hal that translated to the let one period, and the raphs are o cos. For a cos, the amplitude is a. indeed identical. 8. True. Since cos sin, cos sin, and so is a relection in the -ais o sin. 8. Answers will var. 85. Since the raphs are the same, the conjecture is that = sin cos. 8. sin, cos 0 sin = Conjecture: sin cos cos 0 0 0

13 Section.5 Graphs o Sine and Cosine Functions (a) () (c) The raphs are nearl the same or < <. sin 5 7! 5! 7! cos!!! The raphs now aree over a wider rane, < <. The raphs are nearl the same or < <. 88. (a) sin ! 5! () sin! ! (c) sin (e) sin 0.79! sin 0.5 ( calculator) 5 5! ( calculator) cos! 0.57! cos 0.50 ( calculator) sin 0.85 ( calculator) (d) cos !! cos ( calculator) () cos!! cos ( calculator) The error in the approimation is not the same in each case. The error appears to increase as moves arther awa rom lo 0 lo 0 lo lo lo lo lo lo t 9. ln t ln t lnt ln t lnt 9. ln z z ln z z ln z lnz ln z lnz 9. lo 0 lo 0 lo 0 lo 0 9. lo lo lo lo lo () lo 95. ln ln ln ln ln

14 8 Chapter Trionometr 9. ln ln ln ln ln ln ln ln ln ln ln ln 97. Answers will var. Section. Graphs o Other Trionometric Functions You should e ale to raph a tan c a cot c a sec c a csc c When raphin a sec c or a csc c ou should irst raph a cos c or a sin c ecause (a) The -intercepts o sine and cosine are the vertical asmptotes o cosecant and secant. () The maimums o sine and cosine are the local minimums o cosecant and secant. (c) The minimums o sine and cosine are the local maimums o cosecant and secant. You should e ale to raph usin a dampin actor. Vocaular Check. vertical. reciprocal. dampin. 5. n.,, 7.. sec. tan. cot Matches raph (e). Asmptotes:, Matches raph (a). Matches raph (c).. csc 5. Matches raph (d). sec Asmptotes:, Matches raph ().. sec Asmptotes:, Relected in -ais Matches raph ().

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