Math 0 Unit 8: SINUSODIAL FUNCTIONS NAME: Sectin 8.: Understanding Angles p. 8 Hw can we measure things? Eamples: Length - meters (m) r ards (d.) Temperature - degrees Celsius ( C) r Fahrenheit (F) Hw can we measure angles? Up until nw we can measure angles using degrees. There is an alternative unit f measurement fr measuring angles, that is, RADIANS. RADIAN: - One radian is the angle made b taking the radius and wrapping it alng the edge (an arc) f the circle. INVESTIGATE: r radian ~ Degrees r Hw man pieces f this length d u think it wuld take t represent ne cmplete circumference f the circle? T help, cut a piece f pipe cleaner/string t a length equal t radius CA and bend it arund the circle starting at pint A.. Apprimatel, hw man radius lengths are there in ne cmplete circumference?. Hw man degrees in ne cmplete circumference?. Therefre, apprimatel, hw degrees are in ne radian?
NOTE:. The size f the radius f a circle has NO effect n the size f radian.. The advantage f radians is that it is directl related t the radius f the circle. This means that the units f the and ais is cnsistent and the graph f the sine curve will have its true shape, withut vertical eaggeratin. Let s cnsider a circle with radius unit (r = ). (We refer t this as the unit circle!). What is the circumference f a circle?. What is the circumference f the unit circle?. Hw man degrees are there in a cmplete revlutin f a circle?. Wh must the tw equatins be equal t each ther? 5. State the prprtin: 6. Divide bth sides b π radians t determine the measure f radian: 7. Cmplete the fllwing and add the equivalent radian measures n the unit circle belw: a. = b. π = c. d. e. f. 6
Cnverting Degrees Radians. Degrees t Radians: T cnvert frm Degrees t Radians multipl b 80. Radians t Degrees: Eamples: T cnvert frm Radians t Degrees multipl b 80. Cnvert t radians: a. 0 b. 0 c. 50 d. 660. Cnvert t degrees: a. b. 6 c. 9 d. 5. rad. Fr each pair f angle measures, determine which is greater: a. 50, b. 50, 7 c., 8 ASSIGN: p. 89, #,, 5, 7-0
Sectin 8.: Eplring Graphs f Peridic Functins p. 9 Terms t Knw: Peridic Functin A functin whse graph repeats in regular intervals r ccles. Midline / Sinusdial Ais The hrizntal line halfwa between the maimum and minimum values f a peridic functin. Amplitude The distance frm the midline t either the maimum r minimum value f a peridic functin; the amplitude is alwas epressed as a psitive number. Perid The length f the interval f the dmain t cmplete ne ccle. Sinusdial Functin An peridic functin whse graph has the same shape as that f = sin.
5. Cmplete the table f values belw fr the functin sin 0 0 5 60 90 0 5 50 80 5 70 5 60 90 0 50 80 50 50 585 60 675 70. Sketch the graph f sin n the graph belw: 90 80 70 60 50 50 60 70 - -. Cmplete the tables belw b using the graph. If u wanted t quickl graph the sine curve, which five pints wuld allw u t easil graph the entire curve? Perid Sinusidal Ais (midline) Amplitude Five Ke Pints Dmain Range Lcal Maimums Lcal Minimums -intercepts -intercepts
6. Cmplete the table f values belw fr the functin cs 0 0 5 60 90 0 5 50 80 5 70 5 60 90 0 50 80 50 50 585 60 675 70. Sketch the graph f cs n the graph belw: 90 80 70 60 50 50 60 70 - -. Cmplete the tables belw b using the graph. If u wanted t quickl graph the csine curve, which five pints wuld allw u t easil graph the entire curve? Perid Five Ke Pints Sinusidal Ais (midline) Amplitude Dmain Range Lcal Maimums Lcal Minimums -intercepts -intercepts
7 OBSERVATIONS: Fr SINE, ne cmplete wave can be seen frm an X-INTERCEPT at (0, 0) t the X-INTERCEPT at (60, 0). Fr COSINE, ne cmplete wave can be seen frm the MAXIMUM pint (0, ) t the net MAXIMUM pint at (60, ). The graph f cs is related t the graph f sin b a shift f 90 t the left. Label the fllwing as peridic, sinusidal r bth. a. c. b. d. CONCLUSION: All sinusidal functins are peridic but nt all peridic functins are sinusidal. ASSIGN: p. 9, #, 5-8
8 Sectin 8.: The Graphs f Sinusidal Functins p. 97 Eamples:. Fr the sinusidal functin shwn, determine: (Eample, p. 99) a. Range: b. Equatin f Midline/ Sinusidal Ais: c. Amplitude: d. Perid:. While riding a Ferris wheel, Masn s height abve the grund in terms f time can be represented b the fllwing graph. a. Hw far is the ferris wheel ff the grund? b. What is the range f the functin? What des it represent? c. What is the height f the ferris wheel?
9 d. What is the equatin f the midline? What des it represent? e. Hw lng des it take fr the ferris wheel t make ne cmplete revlutin? What characteristic des this crrespnd t?. Aleis and Clin wn a car and a pickup truck. The nticed that the dmeters f the tw vehicles gave different values fr the same distance. As part f their investigatin int the cause, the put a chalk mark n the uter edge f a tire n each vehicle. The fllwing graphs shw the height f the tires as the rtated while the vehicles were driven at the same slw, cnstant speed. What can u determine abut the characteristics f the tires frm these graphs? (Eample, p. 50) ASSIGN: p. 507, #, 5, 7 0, - 5
0 Sectin 8.: The Equatins f Sinusidal Functins p. 56 INVESTIGATION: Using technlg, we will eplre hw the parameters a, b, c and d affect the graph f sinusidal functins written in the frm: asin b( c) d and acs b( c) d (A) The Effect f a in = asin n the graph = sin where a > 0. Sketch a graph f: sin, sin, sin, 0.5sin sin sin -60-70 -80-90 90 80 70 60-60 -70-80 -90 90 80 70 60 - - - - - - - - sin 0.5sin -60-70 -80-90 90 80 70 60-60 -70-80 -90 90 80 70 60 - - - - - - - -. Cmpare the amplitudes in each graph with its equatin.. Describe the affect the value f a has n the graph f sin.. Will the value f a affect the csine graph in the same wa that it affects the sine graph?
(B) The Effect f d in = sin + d n the graph = sin 5. Sketch a graph f: sin and sin and cmpare it t the graph f sin sin sin -60-70 -80-90 90 80 70 60-60 -70-80 -90 90 80 70 60 - - - - - - - - 6. Hw des each graph change when cmpared t sin? 7. Hw is the value f d related t the equatin f the midline? 8. Is the shape f the graph r the lcatin f the graph affected b the parameter d? 9. Is the perid affected b changing the value f d? 0. Will the value f d affect the csine graph in the same wa that it affects the sine graph?
(C) The Effect f b in = sinb n the graph = sin. Sketch a graph f: sin and sin0.5and cmpare it t the graph f sin sin sin0.5-60 -70-80 -90 90 80 70 60-60 -70-80 -90 90 80 70 60 - - - -. What is the perid f sin? What is the b value?. What is the perid f sin? What is the b value?. What is the perid f sin0.5? What is the b value? 5. What is affected b the value f b? 6. Write an equatin that relates the b value t the perid f the functin. Perid r Perid 7. Will the value f b affect the csine graph in the same wa that it affects the sine graph?
(D) The Effect f c in = sin( c) n the graph = sin 8. Sketch a graph f: sin( 60 ) and sin( 0 ) and cmpare it t the graph f sin sin( 60 ) 0 sin( 0 ) sin( ( 0 )) maimum maimum -80-90 90 80 70 60-60 -70-80 -90 90 80 70 60 minimum - minimum - - - 9. Hw is the c value affecting the graph? This hrizntal shift is als called the phase shift f the graph. In rder t determine the phase shift f the graph, u need t cmpare a KEY POINT n the sine graph t determine if it has shifted left r right. One ke pint n the SINE graph is the pint (0, 0) which intersects the midline at = 0 ging frm a minimum pint t a maimum. 0. If the c value is psitive, the graph shifts t the.. If the c value is negative, the graph shifts t the.. Will the value f c affect the csine graph in the same wa that it affects the sine graph?
(E) The Effect f c in = cs( c) n the graph = cs. Sketch a graph f: cs( 90 ) and cs( 60 ) and cmpare it t the graph f sin cs( 90 ) 0 cs( 60 ) cs( ( 60 )) maimum maimum -80-90 90 80 70 60-60 -70-80 -90 90 80 70 60 - - - - One KEY pint n the COSINE graph is the pint (0, ) which is a maimum pint.. If the c value is psitive, the graph shifts t the. 5. If the c value is negative, the graph shifts t the. CONCLUSION: Fr the sinusidal functins written in the frm: asin b( c) d and acs b( c) d. a = amplitude = ma min. b affects the perid: 60 perid r b perid b. c = hrizntal shift (need t cmpare a Ke pint). d = vertical translatin, = d = ma min = equatin f the midline/sinusidal ais 5. maimum value = d + a; minimum value = d a
5 NOTE: Beware f the brackets! cs( ) versus cs cs( ) results in a hrizntal translatin cs results in a vertical translatin Eamples:. Fr the functin, cs, state: (Eample, p. 58) a. amplitude: b. equatin f the midline: c. range: d. perid: e. phase shift:. Fr the functin, sin ( 5 ), state: (Eample, p.59) a. amplitude: b. equatin f the midline: c. range: d. perid: e. phase shift: f. Hw wuld the graph f cs( 5 ) be the same? Hw wuld it be different?
6. Match each graph with the crrespnding equatin belw. (Eample, p. 5) i. cs( 90 ) iii. 5sin( 60 ) ii. sin( 60 ) iv. cs( 60 ). Ashle created the fllwing graph fr the equatin sin( 90 ). Identif her errr(s) and cnstruct the crrect graph. 5 5 80 60 50 70 80 60 50 70 - - - - - - - - -5-5 5. The temperature f an air-cnditined hme n a ht da can be mdeled using the functin t( ).5(cs5 ) 0, where is the time in minutes after the air cnditiner turns n and t() is the temperature in degrees Celsius. a. What are the maimum and minimum temperatures in the hme? b. What is the temperature 0 minutes after the air cnditiner has been turned n? c. What is the perid f the functin? Interpret this value in this cntet? ASSIGN: p. 58, # 5, 7 9
7 Sectin 8.5: Sinusidal Regressin p. 56 As in previus units, we can mdel data that fllws a sinusidal pattern b using the regressin feature n a graphing calculatr. IMPORTANT NOTE: When cmpleting a sinusidal regressin, ur calculatr shuld be in RADIAN mde! Eample : Meterlgist Bran Ndden recrded the average precipitatin fr Grand Falls Windsr fr 0 and created a sinusidal regressin fr the data. ( represents the number f mnths.) A) Use the graph t predict the amunt f precipitatin in August 0. B) Write the equatin f the curve f best fit. C) Use the equatin t predict the amunt f precipitatin in March 07. ASSIGN: p. 5, #,, 5, 7, 8, 0, +Wrksheet