Circular Functions (Trigonometry)

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Julianna Walsh
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2 Circular Fuctios (Trigoometry) Circular fuctios Revisio Where do si cos ad ta come from? Uit circle (of radius ) cos is the coordiate si is the y coordiate si ta cos all are measures of legth. Remember SOH CAH TOA Eact values: 0 o o 0 cos si 0 0 o 90 o 80 o 70 o 0 o ta 0 udefied 0 udefied 0 Agle coversios (betwee radias ad degrees).

3 F X D r a w Quadrats ad symmetry: o All Studets Talk C.. (ASTC) S A T C Fidig Eact values: Eample: (a) What is the eact value of: si ; (b) ta. (a) (b). Sig: rd Quadrat -ve. Agle Equivalet ( st Quadrat):. So: si si or. Sig: d egative Quadrat +ve. Agle Equivalet ( st Quadrat):. So: ta ta Jump Start Holiday Questios Review: radias defiitios eact values symmetry EA Q (ace for all); EB Q acegik acegikmoqsu aceg abdfgj EC Q *CALCULATOR MODE: Always work i radias*

4 Solvig equatios ivolvig circular fuctios. Fidig ais itercepts:. Y-itercepts: f (0) or 0. E.g. what is the Y-itercept of f ( ) si f (0) si 0 f (0) si o f (0) si f (0) f (0). X-itercepts: f ( ) 0 or y 0. Eamples: Fid all values of for: (a) : cos 0. Sig: (+ve) Q. Agle:. (b) : si Sig: (-ve) Q. Agle: si (0.7) (c) : si 0 Rewrite: si. Sig: (-ve) Q - -. Agle:

5 . 7 0 (d) 0 0 cos Rewrite: cos Let 0 a a cos a. Sig: (-ve) Q 7. Agle:. 0 8 a a EE ace ac ac ab abc ace 7 ace 8 acegi; EJ 0 Eam Questio

7 Graphs of Circular Fuctios si y y y cos y Period = Amplitude = Rage: [- ] period - y ta y Period = We do t refer to the amplitude for y ta Rage: R period

c: a traslatio of c uits alog the y-ais.

8 Trasformatios of y si & y cos y si y asi b c & cos y y acos b c a: a dilatio of factor a from the -ais. : a dilatio of factor from the y-ais. b: a traslatio of b uits alog the -ais. c: a traslatio of c uits alog the y-ais.. Dilatios (a) The effect of a Graph the followig graphs: (i) (i) y cos ; (ii) si (ii) y ; where 0 a affects the amplitude. (b) The effect of Graph the followig graphs: (i) (i) y cos ; (ii) si (ii) y ; where 0 affects the period. period

9 . Reflectios. Two types: o Reflectio i the -ais: f () o Reflectio i the y-ais: f ( ) Eamples: Sketch the graphs of the followig: (a) y si ; (b) y cos ; where 0 (a) y (b) y Traslatios (a) The effect of c Sketch the followig: (i) y si ; (ii) cos (i) (ii) y ; where 0 y

10 (b) The effect of b Sketch the followig: (i) y si ; (ii) cos (i) Combiig all trasformatios y ; where 0 Eample: Sketch the graph of f ( ) si [0 ] Rewrite: f ( ) si a b c ad Sketch y f ( ) si first: Secodly with traslatios: (ii) y y y Note: X-itercepts eed to be foud!! EF adfhi ; EG ac ef acfgh 7

11 Graphs &Trasformatios of the Taget fuctio Eample: Sketch y ta for Rewrite: y ta y EJ 7 8 9

12 Additio of ordiates (add the y values) Eample: (a) O the same set of aes sketch f ( ) si ad g( ) cos for 0 ; (b) Use additio of ordiates to sketch the graph of y si cos. Note: For y si( ) cos( ) it is easier to do y si( ) ( cos( )) EH ace

13 Solvig Equatios where both si & cos appear Eample: Solve for 0 si 0.cos divide both sides by cos si 0.cos (i) cos cos ta 0. :. Sig: (+ve) Q. Agle: ta (0.) (ii) si cos 0 si cos 0 ta 0 ta EJ 0 acegi

14 Geeral Solutios to Circular Fuctios Eample: Solve cos Solutio: Z Check geerally Agle Cos 0 : : : Quad...- positive. cos or So i geeral terms: Z a a ) ( cos cos Eample: Solve si Solutio: 0 : ) ( 0 : : : Quad...- positive. si Check or Z Check geerally Agle Si So i geeral terms: Z a or a a ) ( si ) ( ) ( si si The above ca be simplified to Z a ) ( si ) ( For Z a a ) ( ta ta

15 Eample : Fid the geeral solutio for si Solutio: ) ( 7 ) ( 7 7 ) ( 7 si ) ( si si si ) ( si ) ( ) ( si or or or or a or a Eample : Fid the geeral solutio to cos ad hece fid all the solutios from. Solutio: cos cos cos ) ( cos - i domai)& (ot - & - & - - & - - i domai) (ot - - i domai)& (ot domai of solutios for solutio geeral or or a EK ab 8 9

17 Determiig Rules for Circular Fuctios Eample: The graph show has the rule of the form: y y acos ( t b) c fid a b c & a : rage a c : middle of rage y c : period b : from previous kowledge : b or 0 is o the curve : cos(0 b) cosb cosb b b EI 7 8 9; EJ

18 Applicatios of Circular Fuctios Rewrite: ( t ) T 7 cos a 7 b c period hours at t 0 T (0 ) 7 7cos 7cos The machie will ot be able to operate for hours i.e 0 t. (i.e from 0a.m. to 0 p.m.) EL E N

19 Past Eam Questios

21 00

23 0

25 0

27 0

30 0

31 0

CALCULUS BASIC SUMMER REVIEW

CALCULUS BASIC SUMMER REVIEW NAME rise y y y Slope of a o vertical lie: m ru Poit Slope Equatio: y y m( ) The slope is m ad a poit o your lie is, ). ( y Slope-Itercept Equatio: y m b slope= m y-itercept=