Corrections for the textbook answers: Sec 6.1 #8h)covert angle to a positive by adding period #9b) # rad/sec

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1 U n i t 6 AdvF Date: Name: Trignmetric Functins Unit 6 Tentative TEST date Big idea/learning Gals In this unit yu will study trignmetric functins frm grade, hwever everything will be dne in radian measure. Just like length can be measured in meters r yards, angles can be measured using different units f measure. Radian measure, as yu will learn, actually has n units at all. Yu will review primary and secndary trignmetric ratis, cntrast reciprcal trig functins with inverse trig functin, and graph these functins with and withut transfrmatins. Yu will nce again use the unit circle t find slutins f the angles as well as use special triangles t find exact values fr ratis all f this in radian measure nw. Finally yu will again apply the knwledge f sinusidal functins f grade t slve real life wrd prblems as well as review rates f change with trignmetric functins. Crrectins fr the textbk answers: Sec 6. #8h)cvert angle t a psitive by adding perid #9b) 8. # rad/sec Sec 6. #b) n neg n 6 #9b) 0 beats/min Sec 6.6 #0a) y=.7cs[pi/6(x-7)]+ b) 9.6 # using mnths y=6.cs(pi/(x-7))+. where x= is January Sec 6.7 #9a)shift fr csine b) fast at, 6, 8, slw at 0,, Review #6 y= 0sin[pi/(t-0.)]+0 Success Criteria I understand the new tpics fr this unit if I can d the practice questins in the textbk/handuts Date pg Tpics # f quest. dne? Yu may be asked t shw them - Review Radicals and Unit Circle TWO Handuts -7 Radian Measure days Sectin 6. & THREE Handuts 8-9 Exact Values & Other Questins Sectin 6. (SKIP equivalent expressins fr nw cvered in the next unit) & THREE Handuts 0- Sketching Sinusidals Sectin 6. & Handut - Parent Graphs f Trig Functins & Transfrmatins f Trig Functins Sectin 6. & 6. & TWO Handuts 6-7 Wrd Prblems with Sinusidals Sectin 6.6 & TWO Handuts 8-9 Rates f Change Sectin 6.7 REVIEW Questins I had difficulty with ask teacher befre test! Reflect previus TEST mark, Overall mark nw.

2 U n i t 6 AdvF Date: Name: Review Radicals and Unit Circle. Summarize the rules f simplifying radicals: Hw t reduce, hw t add/subtract/multiply/divide and ratinalize. Use the fllwing examples in yur explanatins. a. b c. 6 d. e. 6 g.. Review the fllwing by stating a definitin r by shwing n a diagram: (terminal arm, initial psitin, psitive angles, negative angles, cterminal psitin, standard psitin, principle angle, related acute angle)

3 U n i t 6 AdvF Date: Name:. Explain the signs f primary trig ratis by CAST r by their definitin.( It is imprtant t knw the definitins since CAST desn t wrk fr angles n the x&y axes.). Predict whether each value will be psitive r negative withut using the calculatr. a. tan9 b. sin( ) c. cs670. Find the fllwing ratis withut using the calculatr. a. cs( 90 ) b. cs80 c. tan 70 d. sin Fr the rati sinθ =, the angle θis in standard psitin 0 60 θ. a. Hw many answers fr θ are there? 7. Fr the pint P(,6) a. Sketch the angle, θ, in standard psitin b. Is θ acute, btuse r reflex in Quadrant III r reflex in Quadrant IV? b. Find ctθ c. Find all pssible measures f θ in the given dmain. c. Find the angle θ.

U n i t 6 AdvF Date: Name: Radian Measure Just like when yu measure length, where yu can use centimeters r inches, yu can measure the angles using different systems f measure.

4 U n i t 6 AdvF Date: Name: Radian Measure Just like when yu measure length, where yu can use centimeters r inches, yu can measure the angles using different systems f measure. Yu can use: Degrees, r Revlutins, r Radians Degrees and Revlutins Degrees and revlutins are simple, lk at the fllwing example t see if yu understand these tw systems:. 60 is equivalent t f a revlutin 6 This is because revlutin = 60 S t cnvert 60 yu need t multiply it by the rati f rev 60 rev 60 = rev (degree units will cancel) 60 6 This shuld make sense since there are 6 sectins f 60 that make a full revlutin f revlutin is equivalent t 0 T cnvert rev yu need t multiply it by the rati f 60 rev rev = = 0 (rev will cancel) rev Ntice that in bth examples we multiplied by a rati that is equivalent t ONE 60 rev and rev since rev = NOTE Hw d yu decide which rati t multiply by? Well, yu shuld lk at units. If yu want t get rid f degrees put the degrees in the rati n tp/bttm s that they can cancel get rid f revlutins put the rev n n tp/bttm s that they can cancel. Cnvert t revlutins. Cnvert. revlutins t degrees

5 U n i t 6 AdvF Date: Name: Radians Unlike degrees and revlutins, radian measure desn t have any units at all. It is a rati f lengths. Definitin f radian rati is the fllwing frmula. (It is nt derived frm anything; we define it t be this rati.) arc length angle (in radians)= radius a θ = r Let s figure ut hw t cnvert radians t degrees r revlutins. Cnsider a full revlutin. Arc length will be the full circumference ( C = π r) and the angle in degrees fr a full revlutin is 60. a θ = r π r 60 = (radius cancels) r 60 = π r 80 = π Since revlutin = 60 =π, yu can cnvert frm ne measure t the next by using the fllwing ratis that are equivalent t ONE. NOTE We just fund that π = 80 Hw can π be. AND 80?. and 80 are nt the same numbers and s are nt equal, hwever lk at units. 80 has a degree symbl and. desn t. S π can be replaced with. if yu want t use radians (n units) r 80 when yu want t use degrees. 60 rev rev π π 60 r reciprcals f these (in these ratis use π as.) Ntice that radians will nt have any units. Lk at the fllwing example t see this.. Arc length is.8cm and radius is.cm. a) What is the angle in radians? a θ = r.8cm θ =.cm θ.0 n units!! (centimeters cancel) b) Cnvert t degrees = 7 π π c) Cnvert t revlutins. rev.0rev.0 = 0.8rev π π NOTE Remember radians have n units at all and s if cnverting t radians try t cancel all units. 6. Cnvert t radians 7. Cnvert. revlutins t radians 8. What is the arc length if the angle is 6 and the radius is cm

6 6 U n i t 6 AdvF Date: Name: Linear and Angular Velcities Let s use the fllwing variables: t = time (sec, min, hr) a = θ = v = ω = arc length (mm, cm, m, km) angle (degrees, revlutins, radians) linear velcity/speed (cm/sec, km/hr,...) angular velcity/speed ( /sec, rev/sec, rad/sec, rpm,...) a Angle in radians is θ = by definitin. r a Linear velcity is v = since speed is distance t divided by time (here distance travelled wuld be the arc length) Angular velcity is θ ω = by definitin. t Deriving a frmula fr ω in terms f v and r : rewrite the angle and the linear velcity frmulas a = θ r and a = vt equating these θ r = vt slving fr θ vt θ = r substituting this int the angular velcity frmula vt ( ) Here are all the frmulas that yu will need and all the rati cnversins that yu may need. r vt vt v v ω = = t = = therefre ω = t r r t r r θ = a r θ ω = t v = a t v ω = π 60 r 60 rev rev π 00cm m 000m km 60 sec min 60 min hr 9. Suppse a 0cm diameter wheel is rtating at rpm. At what rate is the wheel mving alng the rad in m/hr? 0. A bicycle has 70cm wheel diameter, hw many rtatins per secnd des the cyclist have t achieve t push the bicycle alng a flat surface at km/hr? 6

7 7 U n i t 6 AdvF Date: Name:. Find the angular velcity in radians/sec f a pint n a water wheel if the wheel makes 00 revlutins in minute.. A large clck has its secnds hand travelling at 6cm/sec. Find the length f the secnd hand.. A plane travels in a circular path very quickly at 60km/hr, n a circle with radius 8m, find the number f rtatins that the plane makes per secnd. 7

8 8 U n i t 6 AdvF Date: Name: Exact Values & Other Questins. What answer is better t recrd f the tw belw, and why? π π cs = r cs = 6 6. Almst everytime trig functins are used there is runding errr. Hwever, it is pssible t find exact values fr sme special angles. Recall the tw special triangles yu have learned in grade, use radians this time.. Yu shuld get cmfrtable at drawing angles given in radian measure. Explain using the fllwing examples hw lking at the denminatr helps yu decide: hw many pieces shuld yu cut pi int. a. 7 π b. π c..0 d. 6 π e. π. Fr each f the questins abve, use the related acute angle t draw the special triangle, if pssible, then fr ne f the questins find all primary trig ratis and fr ne f the questins find the secndary trig ratis. 8

9 9 U n i t 6 AdvF Date: Name:. Fr each f the fllwing draw the terminal arms in crrect quadrants then find all answers fr the pssible angles within first psitive revlutin. (Find exact angles if pssible) a. csθ = b. sinθ = c. sinθ = d. csθ = e. tanθ = f. tanθ = 6. P(, -) frms a principal angle θ, find exact values f secndary trig ratis f θ 7. ctθ = find exact values f primary trig ratis f θ 9

10 0 U n i t 6 AdvF Date: Name: Sketching Sinusidals T sketch transfrmed sine r csine: Yu can als apply the previus technique t sinusidals, but it maybe quicker t d the fllwing Sketch three hrizntal lines and label the values n them (Dn t bther drawing the x and y axes) MAX = c + a MAX MIN amplitude = a = MAX + MIN axis = c = recall that 60 π perid = p = = k k MIN = c a Decide n the shape f the wave and sketch it between the lines yu just drew Label the pints f the cycle as fllws st pint = d last pint = d + perid left pint + right pint middle pint(s) =. Sketch the fllwing a. y = 6sin( π x + π ) + π x b. y = 8cs 0

11 U n i t 6 AdvF Date: Name: c. y = 0.sin( π x). π d. y = cs x 6. Fr each f the fllwing find a few equatins that mdel the graph.

12 U n i t 6 AdvF Date: Name: b) d) e)

13 U n i t 6 AdvF Date: Name: Parent Graphs f Trig Functins. Recall frm the ratinals unit what the relatinship is between characteristics f f ( x ) graph and its reciprcal, trig functins., graph. Use that knwledge t sketch the secndary trig functins vertp f the drawn primary f ( x ) = = f ( x) sin x = = g( x) cs x = = h( x) tan x Recall frm the functins unit what the relatinship is between characteristics f f ( x ) graph and its inverse, functins f ( x), graph. Use that knwledge t sketch the inverse trig functins vertp f the drawn primary trig sin x cs x tan x As yu can see, reciprcals and inverses are nt the same. [ ] f ( x) f ( x) The ntatin fr the inverse can be cnfused with a negative expnent, which DOES make a reciprcal, hence there are ther names yu shuld knw fr the inverse functins. sin x = cs x = tan x =

U n i t 6 AdvF Date: Name: Transfrmatins f Trig Functins (all types except sinusidal). Hw can yu remember the parent shapes and key values?. Summarize hw t sketch any transfrmed trig functin.

14 U n i t 6 AdvF Date: Name: Transfrmatins f Trig Functins (all types except sinusidal). Hw can yu remember the parent shapes and key values?. Summarize hw t sketch any transfrmed trig functin. y = af [ k( x d)] + c Nte that sme handuts nline have cnstants d and c switched. Dn t wrry abut the name f the cnstant s much as the placement f it in the equatin. Find tw different equatins f the fllwing 6. 7.

15 U n i t 6 AdvF Date: Name: 8. Sketch the fllwing functins a. tan x b. csc( π x + π ) + c. ct x d. sec( x π )

16 6 U n i t 6 AdvF Date: Name: Wrd Prblems with Sinusidals. The pistn in the engine f a small aircraft mves hrizntally relative t the crankshaft, frm a minimum distance f cm t a maximum distance f 7cm. During nrmal cruise pwer settings, the pistn cmpletes 00 rpm (revlutins per minute). a. Sketch the hrizntal distance psitin, h, in centimeters, f the pistn as a functin f time, t, in secnds. b. What is the equatin that mdels this?. Cmmercial bttling machines ften use a circular drum as part f a mechanism t install tps n bttles. One such machine has a diameter f 0 cm, and makes a cmplete turn every sec. A sensr at the left side f the drum mnitrs its mvement. Take the sensr psitin as zer. a. Sketch the graph f the hrizntal psitin f a pint n the drum, h, in centimeters, as a functin f time, t, in secnds. b. What is the equatin that mdels this? c. At what times in the first 0 secnd interval is a pint n the drum 0 cm away hrizntally frm the sensr? d. What is the speed f the rtating sensr in cm/sec? 6

17 7 U n i t 6 AdvF Date: Name:. A buy bbs up and dwn in the lake. The distance between the highest and lwest pints is. m. It takes 6 secnds fr the buy t mve frm its highest pint t its lwest pint and back t its highest pint. Suppse the depth f the water is 7m. a. Sketch the vertical displacement, v, in meters, f the buy as a functin f time, t, in secnds. Assume that the buy is at its equilibrium pint at t = 0 sec and that the buy is n its way dwn at that time. b. What is the equatin that mdels this?. A salespersn selling a car alarm reprts that the sund has a minimum frequency f 0 Hz, the maximum being 0 Hz, and the frequency at t=0 being 700 Hz. The salespersn reprts that the car alarm reaches its maximum frequency after secnd and that the frequency increases befre it decreases. a. Sketch the graph f the frequency, f, in Hertz, as a functin f time, t, in secnds. b. What is the equatin that mdels this? c. What is the frequency f the alarm at. secnds? d. At what times in the first 7 secnds, is the frequency at 000 Hz? 7

18 8 U n i t 6 AdvF Date: Name: Rates f Change. Explain why it is nt pssible t find the exact instantaneus rate f change f trig functins by using the difference qutient. x π. Fr y = cs + 60 find the average rate n x π. Use special triangles t keep answer exact. 8

19 9 U n i t 6 AdvF Date: Name:. A mass n a spring is pulled tward the flr and released, causing it mve up and dwn. Its height, h, in centimeters, abve the flr after t secnds is given by the functin h( t) = 0sin(π t +. π ) +, 0 t a. Sketch the graph f height versus time. b. What des the rate f change f this functin represent? c. Predict fr which tw pints the instantaneus rate be zer. d. Predict fr which tw intervals is the average rate psitive n ne and negative n the ther but same value therwise. e. What is the speed f the spring in cm/s?. A car mving at 60km/hr has wheels f radius 0cm. Mdel the height f a pint n this wheel as a functin f time, assuming the pint starts at minimum height. a. Graphical mdel b. Algebraic mdel 9

Trigonometric Ratios Unit 5 Tentative TEST date

Trigonometric Ratios Unit 5 Tentative TEST date 1 U n i t 5 11U Date: Name: Trignmetric Ratis Unit 5 Tentative TEST date Big idea/learning Gals In this unit yu will extend yur knwledge f SOH CAH TOA t wrk with btuse and reflex angles. This extensin