Be able to sketch a function defined parametrically. (by hand and by calculator)

Transcription

1 Pre Calculus Uni : Parameric and Polar Equaions (7) Te References: Pre Calculus wih Limis; Larson, Hoseler, Edwards. B he end of he uni, ou should be able o complee he problems below. The eacher ma provide addiional maerial during he uni. While no ever assignmen will be colleced or graded, ou will sill be responsible for knowing how o do, eplain, and evaluae ever problem lised. For he duraion of he sud of his uni, ou will need o keep all of he work ha ou accumulae while learning. This includes homework, quizzes, ess, an addiional workshees. Te Secion Objecives Assignmen Trig Review w/s 9.5 Be able o skech a funcion defined paramericall. (b hand and b calculaor) Be able o find he recangular represenaion of a funcion defined paramericall. p. 673 (1 30) Plo poins in he polar coordinae ssem Conver beween polar and recangular coordinaes. Conver beween recangular and polar equaions. p.680 (1 0) p.680 (1 3,39 70) Review w/s

2 Pre Calculus Name: Overview Trigonomer Review Trig Review: I. SOH CAH TOA II. Special Righ Triangles a b III. Eamples: a. Find sin10e b. Find cos300e c. Find an10e IV. Radians (Recall: 180 and ASTC)

3 V. Eamples: 7 a. Find cos 6 b. Find 3 cos c. Find 5 sin 4 d. Find sin 4 e. Given 6cosand sin, find and if: 5 5 i. ii. iii iv. v. vi. 6 4

4 Trigonomer Trig values of Radians Addiional Pracice Find he value of each rig funcion. Draw he angle on a coordinae ssem if necessar sin 4. cos 3 3. sin 4 4. cos sin 3 6. cos 7. 7 an an cos cos an 1. sin an sin cos sin cos an Homework: Workshee

5 Pre Calculus 9.5 Parameric Equaions Name: Parameric curves are relaions (), () for which boh and are defined as funcions of a hird variable,. As in f() and g (). Esseniall, we now will have he abili o no onl ell where an objec is given a poin (, ), bu also when he objec is a ha poin (, ). Given: Wih he Parameer: Skech b hand. Indicae direcion. Find he domain and range. 4 Eample 1: Skech he curve given b he parameric equaions:, for 3

6 Eample : A paricle moving in he coordinae plane in such a wa ha () 3 6and () 4for 0 5. Eample 3: Skech he curve and he orienaion given b () and () for 3 Eample 4: Skech he curve and he orienaion given b () cos and () 6sin for ,,,,,,, T Use a graphing calucaor. Eample 5 : Use a graphing calucaor o graph he curves represened b he parameric equaions. Tell which curves are funcions. a. b. 3 c. 3 d., from 3 3 e. 4sin 3cos

7 Eliminae he parameer. Eample 6: Find he Caresian (recangular) equaions of he curve. Idenif he curve represened b he equaions cos a. b., for 0 4sin 1 c. and 1 1 d. 1 and e. and 1 f. 3 cos and 1 sin

8 9.5 Parameric Equaions Addiional Pracice 1) A paricle moving in he coordinae plane in such a wa ha and 6 for 0 5. Skech he pah of he paricle and indicae he direcion of moion (orienaion) ) A paricle moving in he coordinae plane in such a wa ha and for 3. Skech he pah of he paricle and indicae he direcion of moion (orienaion) 3) A paricle moving in he coordinae plane in such a wa ha 4cos and sin for,, 0,,. Skech he pah of he paricle and indicae he direcion of moion 4 4 (orienaion)

9 4) Use our calculaor o graph he parameric equaions a., 1 b. 4cos, 1 sin 5) A parameric curve is defined is defined b 3, 3. Find he recangular equaion of he curve. Solve he equaion for. 6) A parameric curve is defined is defined b cos, 3sin. Find he recangular equaion of he curve. Wha conic secion does his represen? 7) A parameric curve is defined is defined b 3cos, 3sin. Find he recangular equaion of he curve. Wha conic secion does his represen? 8) A parameric curve is defined is defined b he curve. ln,. Find he recangular equaion of

10 9) A segmen (, ) is deermined b he parameric equaions for 6, 1for 4 5 where represens real numbers. a. Find he resricions on (Domain) b. Find he resricions on (Range) c. Find he recangular equaion of he curve. 10) A segmen (, ) is deermined b he parameric equaions for 33, 4 5 for 3 where represens real numbers. a. Find he resricions on (Domain) b. Find he resricions on (Range) c. Find he recangular equaion of he curve. 11) A segmen (, ) is deermined b he parameric equaions for 1 4 where represens real numbers. 1 53, 4 for a. Find he resricions on (Domain) b. Find he resricions on (Range) c. Find he recangular equaion of he curve.

11 1) A segmen (, ) is deermined b he parameric equaions for 7, 5 1for 1 6 where represens real numbers. a. Find he resricions on (Domain) b. Find he resricions on (Range) c. Find he recangular equaion of he curve. d. Wha is he slope of he line found in c? e. Wha is he inercep of he line found in c? 14) A segmen (, ) is deermined b he parameric equaions for 3 3 where represens real numbers. 1 1, 6 for 3 a. Find he resricions on (Domain) b. Find he resricions on (Range) c. Find he recangular equaion of he curve. d. Wha is he slope of he line found in c? e. Wha is he inercep of he line found in c?

12 Pre Calculus 9.6 Polar Coordinaes Name: I. Polar coordinae ssem Ploing Poins in he Polar Coordinae Ssem Eample 1: Plo A and B on he firs graph, hen C and D on he second graph. 7 A 1, B 5, C 3, D 4, Unlike he Caresian coordinae ssem, in which each poin on he plane is epressed b a unique coordinae pair, each poin in he coordinae plane can be represened b an infinie number of polar coordinae pairs. Eample. A. Find and graph wo polar coordinae pairs ha B. Find and graph polar coordinae pairs ha represens he same poin on he plane as,. represen he same poin on he plane 6 as 3, 3 on he inerval [0, ].

13 Coordinae Conversion Recall: r r r cos sin an Eample 3: Conver each poin o recangular coordinaes. Graph he polar coordinaes. 4 7 a., b. 3, c. ( 3, ) d. 7, e., Eample 4: Recangular o Polar Conversion. Graph he polar coordinaes. a. 1,1 b. 0, c. (4, 4) d. ( 1, 3) Equaion Conversion Eample 5: Describe he graph of each polar equaion and find he corresponding recangular equaion. a. r = b. sin c. r sec 3

14 Eample 6: r acos Eample 7: r asin Eample 6: Graph r 3cos Eample 7: Graph r 5 Eample 6: Graph Eample 7: Graph r 4cos 6

15 9.6 Polar Coordinaes Addiional Pracice Remember: r r r cos sin an 1) Plo A, B, and C on he firs graph, hen D, E, and F on he second graph. 5, 3,3 5, 3, 3 A B C D E, 11 F 3, ) Find wo polar coordinae pairs ha represen he same poin on he plane as graph. 5 3, 4. Then 3) Find a polar coordinae pair ha represen he same poin on he plane as Then graph. 3, on [0, ]. 5) Find a coordinae pair ha represens he same poin on he plane as 4, 6 Then graph. on [0, ]. 6) Find a coordinae pair ha represens he same poin on he plane as Then graph. 7 5, 4 on [0, ]. 7) Find a coordinae pair ha represens he same poin on he plane as 1, 3 Then graph. on [0, ].

16 9) Conver he poin, 3 from polar o recangular coordinaes. Then graph. 10) Conver he poin 5, 4 from polar o recangular coordinaes. Then graph. 11) Conver he poin 3, 4 from polar o recangular coordinaes. Then graph. 1) Conver he poin 1, from polar o recangular coordinaes. Then graph. 13) Conver he poin 5 3, from polar o recangular coordinaes. Then graph. 3

17 15) Conver he poin (, ) from recangular o polar coordinaes. Then graph. 16) Conver he poin 1, 3 from recangular o polar coordinaes. Then graph. 17) Conver he poin 3 3,3from recangular o polar coordinaes. Then graph. 18) Conver he poin ( 1, 0) from recangular o polar coordinaes. Then graph. 0) Creae a rough graph of he 1) Creae a rough graph of he polar curve r 4cos polar curve r ) Creae a rough graph of he 3) Creae a rough graph of he 4 polar curve 3 polar curve r 1 sin