Trigonometric Ratios Unit 5 Tentative TEST date

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1 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 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 learn hw t find the exact ratis fr special angles withut using a calculatr. Yu will then learn the differences between a trignmetric equatin and a trignmetric identity. Yu will practice prving different trignmetric identities. Finally yu will review Sine and Csine laws that yu ve studied in grade 10 and see that the calculatr desn t always give yu the slutin that yu need. Yu will learn hw t interpret the results frm the calculatins and decide n the prper answer. Crrectins fr the textbk answers: Sec 5.6 # m # Sect 5.8 #9 7 Success Criteria I understand the new tpics fr this unit if I can d the practice questins in the textbk/handuts Date pages Tpics # f quest. dne? Yu may be asked t shw them Primary and Secndary Trig Ratis Sectin 5.1 & Handut Obtuse & Reflex Angles n the Unit Circle 1.5 days Sectin 5.3 & Handut Using the Unit circle 1.5 days Sectin 5.4 & Handut Exact Values using Special Angles Sectin 5. & tw Handuts QUIZ n calculatrs Trignmetric Identities & Prfs Sectin 5.5 & Handut Sine Law Review & Ambiguus Case Sectin 5.6 Csine Law Review & Slve 3D Prblems Sectin 5.7 & 5.8 REVIEW days? Questins I had difficulty with ask teacher befre test! Reflect previus TEST mark, Overall mark nw. 1

2 U n i t 5 11U Date: Name: Primary and Secndary Trig Ratis 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:. Slve the fllwing fr X. a. b. c. 3. Frm the tp f a building, the angle f elevatin f the tp f a nearby building is 8 and the angle f depressin f the bttm f the nearby building is 48. The distance between the tw buildings is 50 m. What is the height f the taller building?

3 3 U n i t 5 11U Date: Name: 4. Yu ve been using the primary trignmetric ratis, there are als secndary nes. Summarize what the secndary trignmetric ratis are: 5. Find the fllwing ratis (DON T find angle unless asked) 6. Find the fllwing ratis a. csy b. ctw a. csc A b. tanb 7. If 1 sinθ = 4 and secθ then shw hw t find the values f cscθ 8. If ctθ = then what is θ? 3

4 4 U n i t 5 11U Date: Name: Obtuse & Reflex Angles n the Unit Circle 1. 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: terminal arm initial psitin psitive angle rtatin negative angle rtatin c-terminal angles standard psitin principle angle related acute angle (r reference angle) quadrants acute angles btuse angles reflex angles. State the principal angle and the related acute angle, then state tw mre c-terminal angles. a. b. 4

5 5 U n i t 5 11U Date: Name: 3. 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 0 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? 4. 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? 5

6 6 U n i t 5 11U Date: Name: 5. 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 fllwing ratis: EACH 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? 6. Fr the angle 170 a. Find the related acute r the reference angle 7. 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 and 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. 6

7 7 U n i t 5 11U Date: Name: Using the Unit Circle 1. Predict whether each value will be psitive r negative. Explain the MEANING f each rati. a. tan195 b. sin( 115 ) c. cs670. Fr all f the abve state an equivalent trignmetric expressins with same value f the rati. 3. Find the fllwing ratis withut using the calculatr. a. cs( 90 ) b. cs180 c. tan70 d. sin Find the angles withut using the calculatr. a. csθ = 1 b. csθ = 0 c. sinθ = 1 d. tanθ = undefined 7

8 8 U n i t 5 11U Date: Name: sinθ = 5 psitin i.e. 0 θ Fr the rati, the angle θ is in standard a. Hw many answers fr θ are there? 6. 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. 4 secθ = 3 psitin 0 θ Fr the rati a. Find all 5 ther trig ratis fr θ, the angle θ is in standard 8. 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. 8

9 9 U n i t 5 11U Date: Name: 9. When trig ratis are 0, and sine/csine are ± 1 then: 10. If trig ratis are smething else: 11. Slve fr angle if 0 angle 360 a. = 5tan β b. 4sinθ 3 = 0 c. sin β = 0.5 g. cs t + 1= 0 h. tanθ = 0 i. csα = 0 d. ctω = 5.64 e. sec λ = 1 f.cscϕ = 4 9

10 10 U n i t 5 11U Date: Name: Exact Values using Special Angles 1. What are special angles?. What answer is better t recrd f the tw belw and why? cs30 = r cs30 = 3 3. 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 10

11 11 U n i t 5 11U Date: Name: 6. Find the exact values f each f the fllwing, withut using the calculatr. a. csc330 b. cs70 c.sin315 e. d. tan( 10 ) 0 sec5 f. ct10 0 g.sin70 cs45 ct 60 sec150 h. csc90 3tan135 cs Withut using the calculatr explain hw yu can find the slutins fr angles t the fllwing a. 3 csθ = b. tanθ = 3 3 c. sinθ = 3 d. tanθ = 1 11

12 1 U n i t 5 11U Date: Name: 8. Fr the pint P(, 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. 9. The angle θ, is in Quadrant III, and tw pssible crdinates fr pint P. sinθ = 3. Pint P lies n the terminal arm. Determine θ, and state at least 10. The terminal arm f θ is in quadrant III and n the line 3y 3x = 0. Determine the angle θ in standard psitin 1

13 13 U n i t 5 11U Date: Name: Trignmetric Identities & Prfs 1. Befre yu begin prving identities, it is imprtant t understand the differences between the fllwing wrds. Explain with the use f examples what each term means. EXPRESSION EQUATION IDENTITY. Shw that the fllwing identities are true fr any angle a. sinθ tanθ = csθ b. θ θ sin + cs = 1 3. The prfs t identities are nt unique, hence it is imprtant t write ut what yu d in each step. List sme strategies t try when ding prfs. Things that yu will lse marks fr are: Nt explaning steps r skipping steps Fr writting terms incrrectly eg. cs, sin, tan withut, r sin when yu mean sin θ θ θ Fr incrrectly canceling r simplifying. Mving terms ver the equals sign. This is nt wrng t d, hwever the prfs in grade 11 are nt particularly hard and if terms are mved prfs can becme t simple. 13

14 14 U n i t 5 11U Date: Name: 4. Prve the fllwing a. csθ 1 sinθ + = b. 1 sin θ = tanθ csθ tanθ sin θ tan θ c. sin x + cs x 1 = cs x sin x d. 1 cs x 1 cs x = + e. secθ cscθ ctθ = tanθ f. (sin + cs ) = 1+ sin cs x x x x 14

15 15 U n i t 5 11U Date: Name: Sine Law Review & Ambiguus Case 1. Summarize the sine law and when yu can use it.. Find side BC 3. Find angle X 4. Smetimes slving fr the angles using sine law, the calculatr gives yu the acute angle, when the prblem actually requires an btuse angle, like in ne f the questins abve. Why des this happen? And why dn t yu need t wrry abut this scenari with csine law? 15

16 16 U n i t 5 11U Date: Name: Smetimes when the diagram is nt given an ambiguus case is created. This can happen when yu are given SSA. In this situatin there can be tw pssible triangles that can be slved OR n triangles at all OR nly ne triangle. 5. Determine if there is n triangle t slve at all, r if there is ne triangle r if there are tw triangles that must be slved. a. In ABC A=35, a = 3, b = 4 b. In ABC A=96, a = 13, b = 0 c. In DEF E=8, e = 4, f = 0 6. Find the area f the triangle 7. Same questin as 6, just change angle t 45 and d exact value fr the area. 16

17 17 U n i t 5 11U Date: Name: Csine Law Review & 3D Prblems 1. Summarize the csine law and when yu can use it.. Slve each triangle. This means find all sides, all angles. a. b. 17

18 18 U n i t 5 11U Date: Name: 3. Real life situatins are almst never flat D prblems. Slve the fllwing fr x. a. b. c. d. 18

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