1 PreCalculus AP Unit G Rotational Trig (MCR) Name:
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1 1 PreCalculus AP Unit G Rtatinal Trig (MCR) Name: Big idea In this unit yu will extend yur knwledge f SOH CAH TOA t wrk with btuse and reflex angles. This extensin will invlve the unit circle which will allw yu t understand that inverse sine r inverse csine r inverse tangent actually give mre than ne slutin. Yu will als learn hw t find the exact ratis fr special angles withut using a calculatr. Feedback & Assessment f Yur Success Date Pages Tpics 2days New Terminlgy (MCR) 2-6 Jurnal #1,2 Finished assignment pages? Made crrectins? Summarized ntes in a jurnal? Added yur wn explanatins? Hw many extra practice questins did yu try in each tpic? Tentative TEST date: Questins t ask the teacher: 3days Exact Values using Special Angles (MCR) Jurnal #3 Rtatinal Trignmetry (MCR) Jurnal #4,5 Ratinalize Denminatr 2 5 Ratinalize Numeratr Multiplying 4a 9a x y 3 xy Adding Reducing
2 2 PreCalculus AP Unit G Rtatinal Trig (MCR) Name: New Terminlgy 1. Since yu will be extending yur trignmetry knwledge frm grade 10 t be able t slve btuse and reflex angles, it is a gd idea t review the primary trignmetric ratis, r SOH CAH TOA, and Pythagrean Therem. Summarize what yu shuld knw: 2. Summarize what the secndary trignmetric ratis are: 3. Find the fllwing ratis (DON T find angle unless asked) 4. Find the fllwing ratis a. csy b. ctw a. csc A b. tan B 5. If 1 sinθ = 4 and secθ then shw hw t find the values f cscθ 6. If ctθ = then what is θ? 2
3 3 PreCalculus AP Unit G Rtatinal Trig (MCR) Name: 7. T understand the way btuse and reflex angles relate t the unit circle, yu need t learn sme new definitins. Define(r shw n a diagram) the fllwing terms: standard psitin Initial arm terminal arm Psitive/Negative angle rtatin Quadrants quadrantal angles principle angle c-terminal angles related acute angle (r reference angle) acute angles btuse angles reflex angles 8. State the principal angle and the related acute angle, then state tw mre c-terminal angles. a. b. 3
4 4 PreCalculus AP Unit G Rtatinal Trig (MCR) Name: 9. Yu must understand why smetimes ratis are negative and smetimes psitive. Answer the fllwing questins t see what the pattern is. a. Draw the related acute angle 20 in each quadrant, b. In each quadrant, use the calculatr t find all three primary trig ratis fr the principal angle in that and state the principal angles each frms with the quadrant. initial arm. c. What d yu ntice? d. What des the acrnym CAST stand fr? 10. The acrnym CAST is nt always useful. Angles can fall nt the x r y axes. Answer the fllwing questins t really understand the new definitins f sine, csine and tangent. a. Use the calculatr t find all three primary trig ratis f all the angles that fall nt the axes b. Cmpare yur answers t the crdinate pints that the answers crrespnd t, what d yu ntice? c. What are the new definitins f sine, csine and tangent? (Keep in mind the cicle desn t have t have a radius f 1.) Als, what des the pythagrean therem remind yu f here? 4
5 5 PreCalculus AP Unit G Rtatinal Trig (MCR) Name: 11. Predict whether each value will be psitive r negative. Explain the MEANING f each rati. a. tan195 b. sin( 115 ) c. cs Find the fllwing ratis withut using the calculatr. a. cs( 90 ) b. cs180 c. tan270 d. sin Find the angles withut using the calculatr. a. csθ = 1 b. csθ = 0 c. sinθ = 1 d. tanθ = undefined Determine the a) principal angle, b) related acute angle, c) tw cterminal angles (ne psitive, ne negative)
6 6 PreCalculus AP Unit G Rtatinal Trig (MCR) Name: Find the values f all six trig ratis fr θ 16. in quadrant II 17. nt given a quadrant, s find all pssibilities fr the ratis Predict what quadrant the angle is in Fr all f the fllwing state an equivalent trignmetric expressin with same value f the rati. a. tan195 b. sin( 115 ) c. cs670 6
7 7 PreCalculus AP Unit G Rtatinal Trig (MCR) Name: Exact Values using Special Angles 1. What are special angles? 2. What answer is better t recrd f the tw belw and why? cs30 = r cs30 = Almst everytime trig functins are used there is runding errr. Hwever, it is pssible t find exact values fr sme special angles. Draw tw special triangles and explain where the side lengths cme frm. 4. Des it matter what is the size f the triangle used when dealing with ratis? Draw different sized triangles and label the dimensins. Shw that ratis are still the same. 5. Find the exact values fr all the dimensins f fllwing diagram 7
8 8 PreCalculus AP Unit G Rtatinal Trig (MCR) Name: Find the missing side lengths, leave answers as exact and reduced radicals Slve, leave answers exact Slve, leave answers exact. 12. Find x and y if s=2cm. Hint hexagn is made up f 6 equilateral triangles. 13. Find the missing lengths 8
9 9 PreCalculus AP Unit G Rtatinal Trig (MCR) Name: Rtatinal Trignmetry 1. Yu CANNOT rely n the calculatr t give yu the answers that yu need anymre, t see this, answer the fllwing questins. a. Fr the pint P( 3, 5) find the b. Nw use the inverse buttns fr EACH fllwing ratis: rati t find the angle θ sinθ csθ tanθ c. What d yu ntice with answers the calculatr gives yu fr θ? d. What must be dne when yu use inverse buttns when dealing with btuse r reflex angles? 2. Fr the angle 170 a. Find the related acute r the reference angle 3. Fr tan( 140 ) a. Sketch the given angle b. State the principal angle and the related acute angle b. Predict the signs f all the primary trig ratis. Explain yur chice using CAST r using the new definitins f the trig ratis. c. Find a few equivalent expressins t tan( 140 ) that give the same answer fr the rati. 9
10 10 PreCalculus AP Unit G Rtatinal Trig (MCR) Name: Find ratis 1. Find the exact values f each f the fllwing, withut using the calculatr. a. csc330 b. cs720 c.sin315 e. d. tan( 120 ) 0 sec225 f. ct120 0 g.sin270 cs45 ct 60 sec150 h. 2csc90 3tan135 cs210 10
11 11 PreCalculus AP Unit G Rtatinal Trig (MCR) Name: Find the values f all six trig functins withut using a calculatr Find the rati values, explain hw yu can d s withut using a calculatr
12 12 PreCalculus AP Unit G Rtatinal Trig (MCR) Name: Find equivalent trig rati expressins t the given rati a) using principal angle b) using an angle in quadrant III a) using an angle in quadrant I b) using an angle in quadrant IV Find the new pint n the terminal arm given the angle that the pint rtated thrugh, rund t 2 decimals Find several pssibilities fr pint that lies n the terminal arm then find θ. 12. θ lies n the line 7x + 5y = 0 in quadrant II
13 13 PreCalculus AP Unit G Rtatinal Trig (MCR) Name: Find angles 1. Withut using the calculatr find the slutins fr angles t the fllwing 3 3 a. csθ = b. tanθ = 3 2 c. sinθ = 2 2 d. tanθ = 1 e. secθ = 2 f.ctθ = 3 g. 2 3 cscθ = 3 h. secθ = undefined 13
14 14 PreCalculus AP Unit G Rtatinal Trig (MCR) Name: 2. Slve fr angle if 0 angle 360 If pssible d nt use the calculatr. a. 2 = 5tan β b. 4sinθ 3 = 0 c. sin β = 0.5 d. 2cs t + 1= 0 e. tanθ = 0 f. csα = 0 g. ct ω = 5.64 h.sec λ = 1 i. cscϕ = 4 14
15 15 PreCalculus AP Unit G Rtatinal Trig (MCR) Name: 2 sinθ = 5 psitin i.e. 0 θ Fr the rati, the angle θ is in standard a. Hw many answers fr θ are there? 4. Fr the pint P( 2,6) 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. 4 secθ = 3 psitin 0 θ Fr the rati a. Find all 5 ther trig ratis fr θ, the angle θ is in standard 6. Fr the pint P(5, 7) a. Find cscθ b. Find all pssible measures f θ in the given dmain b. Find the angle θ. 15
16 16 PreCalculus AP Unit G Rtatinal Trig (MCR) Name: 7. Fr the pint P( 2, 8) find the exact values f the three primary trig ratis fr the principal angle that is made with the terminal arm with pint P n it. 8. The angle θ, is in Quadrant III, and tw pssible crdinates fr pint P. sinθ = 3 2. Pint P lies n the terminal arm. Determine θ, and state at least 9. The terminal arm f θ is in quadrant III and n the line 3y 3x = 0. Determine the angle θ in standard psitin 16
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