MHF 4U Unit 4 Polynomial Functions Outline Day Lesson Title Specific Epectations Transforming Trigonometric Functions B.4,.5, 3. Transforming Sinusoidal Functions B.4,.5, 3. 3 Transforming Sinusoidal Functions - continued B.4,.5, 3. 4 Writing an Equation of a Trigonometric Function B.6, 3. 5 Real World Applications of Sinusoidal Functions B.7, 3. 6 Real World Applications of Sinusoidal Functions Day B.7, 3. 7 Compound Angle Formulae B 3.. 3. 8 (Lesson included) 9 (Lesson included) 0 (Lesson included) Proving Trigonometric Identities B3.3 Solving Linear Trigonometric Equations B3.4 Solving Quadratic Trigonometric Equations B3.4 - JAZZ DAY 3 SUMMATIVE ASSESSMENT TOTAL DAYS: 3 Advanced Functions: MHF4U -: Unit 4 Trigonometric Functions II (Draft August 007) Page of 7
Unit 4: Day 8: Proving Trigonometric Identities Learning Goals: Minds On: 0 Demonstrate an understanding that an identity holds true for any value of the independent variable (graph left side and right side of the equation as Action: 55 functions and compare) Apply a variety of techniques to prove identities MHF4U Materials BLM 4.8. BLM 4.8. BLM 4.8.3 BLM 4.8.4 Consolidate:0 Total75 min Minds On Whole Class Investigation Using BLM 4.8. the teacher introduces the idea of proof trying to show something, but following a set of rules by doing it. Action! Whole Class Discussion The teacher introduces the students to the idea of trigonometric proofs (using the trickledown puzzle as inspiration. The teacher goes through several eamples with students. Assessment Opportunities The trickledown puzzle has two rules: you may only change one letter at a time, and each change must still result in a rule. Trig proofs are similar: you must use only valid substitutions and you must only deal with one side at a time. Consolidate Debrief Small Groups Activity Using BLM 4.8.3 students perform the complete the proof activity. The Labels go on envelopes and inside each envelope students get a cut-up version of the proof which they can put in order. When they are finished they can trade with another group. Also, this could be done individually, or as a kind of race/competition. Eploration Application Home Activity or Further Classroom Consolidation Complete BLM 4.8.4 A-W McG-HR H A-W H McG-HR (MCT) 5.9 5.7 9. Appendi p.390-395 Advanced Functions: MHF4U -: Unit 4 Trigonometric Functions II (Draft August 007) Page of 7
4.8. Proving Trigonometric Identities (Teacher Notes) To prove an identity, the RHS and LHS should be dealt with separately. In general there are certain rules or guidelines to help:. Use algebra or previous identities to transform one side to another.. Write the entire equation in terms of one trig function. 3. Epress everything in terms of sine and ines 4. Transform both LHS and RHS to the same epression, thus proving the identity. Known identities: quotient identities reciprocal identities Pythagorean identities Compound Angle Formulae Eample : cot sin LS cot sin sin sin sin sin RS Eample : ( )(csc ) sin LS ( )(csc ) sin csc sin sin sin sin RS sin Advanced Functions: MHF4U -: Unit 4 Trigonometric Functions II (Draft August 007) Page 3 of 7
4.8. Proving Trigonometric Identities (Teacher Notes continued) Eample 3: ( + sec )/ (tan + sin ) csc + sec LS tan + sin + sin + sin + sin + sin + sin + sin + sin + ( ) + sin + sin csc RS ( ) Eample 4: y ( + y) + ( y) RS ( + y) + ( y) y sin ysin + y+ sin ysin y+ y y LS Advanced Functions: MHF4U -: Unit 4 Trigonometric Functions II (Draft August 007) Page 4 of 7
4.8. The Trickledown Puzzle You goal is to change the top word into the bottom word in the space allowed. The trickledown puzzle has two simple rules:. You may only change one letter at a time;. Each new line must make a new word. COAT VASE PLUG STAY SLANG TWINE Proving The Trigonometric Identity Your goal is to show that the two sides of the equation are equal. You may only do this by:. Substituting valid identities. Working with each side of the equation separately [ + ()][ ()] sin () Advanced Functions: MHF4U -: Unit 4 Trigonometric Functions II (Draft August 007) Page 5 of 7
4.8.3 Trigonometric Proofs! Cut the following labels and place each one on an envelope. Label: Label: sin + tan (cot + tan ) sec Label: (+ sin )( sin ) sec Label: sin (csc + sec ) sec Label: tan (cot + tan ) sec Label: tan Label: csc sec cot + tan Label: + tan Label: tan sin Label: sin (csc sin ) Advanced Functions: MHF4U -: Unit 4 Trigonometric Functions II (Draft August 007) Page 6 of 7
4.8.3 Trigonometric Proofs! (Continued) For each of the following proofs cut each line of the proof into a separate slip of paper. Place all the strips for a proof in the envelope with the appropriate label. sin + + (+ )( ) + ( ) + + LS RS tan (cot + tan ) sec tan tan sec tan + + tan sec sin + se sin + c sec sec sec sec LS RS Advanced Functions: MHF4U -: Unit 4 Trigonometric Functions II (Draft August 007) Page 7 of 7
4.8.3 Trigonometric Proofs! (Continued) (+ sin )( sin ) sec sin sec sec sec sec LS RS sin (csc + sec ) sec sin csc + sin sec sec sin + sin sin sin + se sin + sec sec sec c sec sec LS RS + tan sin + + sin LS RS tan ( ) tan sin sin ( ) sin LS RS Advanced Functions: MHF4U -: Unit 4 Trigonometric Functions II (Draft August 007) Page 8 of 7
4.8.3 Trigonometric Proofs! (Continued) csc sec cot + tan sin + sin csc sec sin + csc sec sin sin sin csc sec sin csc sec csc sec csc sec LS RS tan (cot + tan ) sec tan tan sec tan + + tan sec sin + sec sin + sec sec sec sec LS RS tan sin sin tan sin tan sin sin sin tan sin tan sin LS RS sin (csc sin ) sin csc sin sin sin sin sin LS RS Advanced Functions: MHF4U -: Unit 4 Trigonometric Functions II (Draft August 007) Page 9 of 7
4.8.4 Proving Trigonometric Identities: Practice Knowledge. Prove the following identities: (a) tan sin (b) sec (c) (tan )/(sec ) sin. Prove the identity: (a) sin (cot + ) (tan + ) (b) sin tan -sintan (c) ( )(tan + ) -tan (d) 4 sin 4 sin 3. Prove the identity (a) ( y)/[sin y] cot + tan y (b) sin( + y)/[sin( y)] [tan + tan y]/[tan tan y] 4. Prove the identity: (a) sec / csc + sin / tan (b) [sec + csc ]/[ + tan ] csc (c) /[csc sin ] sec tan Application 5. Half of a trigonometric identity is given. Graph this half in a viewing window on [, ] and write a conjecture as to what the right side of the identity is. Then prove your conjecture. (a) (sin / [ + ])? (b) (sin + )(sec + csc ) cot? Advanced Functions: MHF4U -: Unit 4 Trigonometric Functions II (Draft August 007) Page 0 of 7
4.8.4 Proving Trigonometric Identities (Continued) 6. Prove the identity: (a) [ sin ] / sec 3 / [ + sin ] (b) tan tan y(cot cot y) tan tan y 7. Prove the identity: cot / [cot ] [cot + ] / cot 8. Prove the identity: ( sin y) / ( y sin ) ( y + sin ) / ( + sin y) Thinking 9. Prove the double angle formulae shown below: sin sin sin tan tan / [ tan ] Hint: + Answers: All identities in # - 9 can be proven.. (a) conjecture: () (b) conjecture: tan() Advanced Functions: MHF4U -: Unit 4 Trigonometric Functions II (Draft August 007) Page of 7
Unit 4: Day 9: Solving Linear Trigonometric Equations Learning Goals: Minds On: 0 Solve linear and quadratic trigonometric equations with and without graphing technology, for real values in the domain from 0 to Action: 45 Make connections between graphical and algebraic solutions MHF4U Materials BLM 4.9. BLM 4.9. BLM 4.. Consolidate:0 Total75 min Minds On Small Groups Activity Put up posters around the class and divides the class into small groups (as many as there are posters) Groups go and write down their thoughts on their assigned poster for 5 minutes, then they are allowed to tour the other posters ( min each) and write new comments. Once the tour is done, the posters are brought to the front and the comments are discussed Assessment Opportunities BLM 4.8. contains teacher notes to review solving trigonometric equations. Action! Consolidate Debrief Eploration Application The Poster titles are: CAST RULE, SOLVE sin 0, SOLVE, SOLVE sin 0.5, SOLVE 0.5, SOLVE tan 0 Whole Class Discussion The teacher reviews solving linear trigonometric equations. The teacher should use different schemes to describe the solution and its meaning (i.e., each student can have a graphing calculator to use and graph equations, the unit circle can be discussed, etc). Whole Class Investigation Teacher reintroduces the Unit Problem again (using BLM 4..) and discusses how to solve the problems involving finding solutions to the equation. When is the ball m high? etc. This reviews solving an absolute value equation as well. Home Activity or Further Classroom Consolidation Complete BLM 4.9. Process Epectation: Reflecting: Student reflect on the unit problem and all of the ways they have been able to tackle the problem. A-W McG-HR H A-W H McG-HR (MCT) 5.7 5.8 9.3 8., 8. Advanced Functions: MHF4U -: Unit 4 Trigonometric Functions II (Draft August 007) Page of 7
4.9. Trigonometric Equations Review (Teacher Notes) Quick review of CAST rule, special angles, and graphs of primary trig functions Eample : Solve 3 sin, 0 3 sinα 3 α sin 3 + + n, + n 3 3 4 5, 3 3 Eample : Solve, 0 6 6 α α 3 α + n, α + n 6 6 + n, + n 6 3 6 3 + n, + n 6, 6 Here, α is the reference angle (or related acute angle). Let n belong to the set of integers. Find all values of n such that is in desired interval. Here, α is the reference angle (or related acute angle). Let n belong to the set of integers. Find all values of n such that is in desired interval. Advanced Functions: MHF4U -: Unit 4 Trigonometric Functions II (Draft August 007) Page 3 of 7
4.9. Trigonometric Equations Review (Teacher Notes Continued) Eample 3: Solve tan + 0, 0 tan + tan + tanα α tan () 4 + α + n,+ + α + n 4 + + n,+ + + n 4 4 3 + n, + n 4 4 3 + n, + n 8 8 7 3 5,,, 8 8 8 8 Here, α is the reference angle (or related acute angle). Let n belong to the set of integers. Find all values of n such that is in desired interval. The n-notation is important for students to realize the infinite number of solutions and how we are simply taking the ones that lie in the interval given. Advanced Functions: MHF4U -: Unit 4 Trigonometric Functions II (Draft August 007) Page 4 of 7
4.9. Solving Linear Trigonometric Equations Knowledge. Find the eact solutions: 3 ( a)sin ( b)tan 3 ( c) 3. Find all solutions of each equation: ( a)cot.3 ( b) ( c)53 3 3. Find solutions on the interval [0, ]: ( a) tan 0.37 ( b) csc ( c) sin Application Use the following information for questions 4 and 5. When a beam of light passes from one medium to another (for eample, from air to glass), it changes both its speed and direction. According to Snell s Law of Refraction, sinθ v sinθ v Where v and v are the speeds of light in mediums and, and θ and θ are the angle of incidence and angle of refraction, respectively. The number v /v is called the inde of refraction. 4. The inde of refraction of light passing from air to water is.33. If the angle of incidence is 38, find the angle of refraction. 5. The inde of refraction of light passing from air to dense glass is.66. If the angle of incidence is 4, find the angle of refraction. 6. A weight hanging from a spring is set into motion moving up and down. Its distance d (in cm) above or below its rest position is described by d 5sin6 ( t 46t ) At what times during the first seconds is the weight at the rest position (d 0)? Advanced Functions: MHF4U -: Unit 4 Trigonometric Functions II (Draft August 007) Page 5 of 7
4.9. Solving Linear Trigonometric Equations (continued) 7. When a projectile leaves a starting point at an angle of elevation of θ with a velocity v, the horizontal distance it travels is determined by v d sin θ 3 Where d is measured in feet and v in feet per second. An outfielder throws the ball at a speed of 75 miles per hour to the catcher who it 00 feet away. At what angle of elevation was the ball thrown? 8. Use a trigonometric identity to solve the following: sin + sin 6 6 Thinking 9. Let n be a fied positive integer. Describe all solutions of the equation sin q Answers: 5. ( a) + n or + n ( b) + n ( c) ± + n 3 3 3 6 n. (a) 0.40 + k, (b) ± + 4n (c) ± 0.738+ 3 3 7 3. (a).9089, 6.05048 (b), (c), 4 4 6 6 tan 4 n 4. 7.57 5. 4.8 6. t +.68, 0.7466, 0.0,.798 6 6 7. 6.0 or 74.0 8. sin( + ) 6 7 + n or + n n 5 n 9. +, + 6q q 6q q Advanced Functions: MHF4U -: Unit 4 Trigonometric Functions II (Draft August 007) Page 6 of 7
Unit 4: Day 0: Solving Quadratic Trigonometric Equations Learning Goal: Minds On: 0 Solve linear and quadratic trigonometric equations with and without graphing technology, for real values in the domain from 0 to Action: 55 Make connections between graphical and algebraic solutions Consolidate:0 MHF4U Materials BLM 4.0.) BLM 4.0. BLM 4.0.3 Graphing Calculators Total75 min Assessment Opportunities Minds On Pairs Matching Activity Using BLM 4.0. the teacher gives pairs of students cards on which they are to find similar statements. (they are trying to match a trigonometric equation with a polynomial equation) On their card they must justify why their equations are similar. Teacher can give hints about the validity of their reasons. The reasons for the matches are then discussed using an overhead copy of BLM 4.0. (Answers) Action! Individual Students Discussion/Investigation The teacher goes through solutions to the different cards. Then using that as a basis students work through BLM 4.0. to develop strategies and skills in solving quadratic trigonometric equations. Consolidate Debrief Small Groups Graphing Calculators Students are then put into small groups to discuss their answers to 4.0. and to use graphing calculators to see the graphs of the functions and determine if their solutions are correct. Home Activity or Further Classroom Consolidation Eploration Application Complete BLM 4.0.3 A-W McG-HR H A-W H McG-HR (MCT) 5.7 5.8 9.3 8., 8. Advanced Functions: MHF4U -: Unit 4 Trigonometric Functions II (Draft August 007) Page 7 of 7
4.0. Trigonometric Equations: Matching Cards sin 0 tan tan 0 + 3+ 0 + 3 + 0 y + y 0 sin + sin 0 y cot cot y 0 sin 0 Advanced Functions: MHF4U -: Unit 4 Trigonometric Functions II (Draft August 007) Page 8 of 7
4.0. Trigonometric Equations: Matching (Solutions) Use the following notes to help eplain the process of solving quadratic trig equations (and their similarity to polynomial equations) ± ± 0 ( ) 0 0 or sin sin ± sin ± n + 4 equation has solutions in all quadrants tan tan 0 tan (tan ) 0 tan 0 n or or tan + n 4 equation has solution on -ais, or on y Advanced Functions: MHF4U -: Unit 4 Trigonometric Functions II (Draft August 007) Page 9 of 7
4.0. Trigonometric Equations: Matching (Teacher Notes) + 3+ 0 ( + )( + ) 0 or + 3 + 0 ( + )( + ) 0 or ( no solution) or + n y+ y 0 y( + ) 0 y 0 or + 0 y y 0 y ( ) 0 y 0 or 0 y sin + sin 0 sin ( + ) 0 sin 0 or + 0 n or 4 + n, + n 3 3 cot cot cot cot 0 cot ( ) 0 cot 0 or 0 + n or n Advanced Functions: MHF4U -: Unit 4 Trigonometric Functions II (Draft August 007) Page 0 of 7
4.0. Trigonometric Equations: Matching (Teacher Notes) y 0 sin 0 0 or y 0 0 or sin 0 + n or n Advanced Functions: MHF4U -: Unit 4 Trigonometric Functions II (Draft August 007) Page of 7
4.0. Solving Quadratic Trigonometric Equations In the following 4 eamples, you will deal with 4 different methods to deal with quadratic trigonometric equations. Method : Common Factor Solve for θ,0 θ θ θ θ tan tan Move all terms to LS and factor out the common factor. Method : Trinomial Factor Solve for,0 3sin sin 0 Replace sin() with q Can you factor 3q q? Advanced Functions: MHF4U -: Unit 4 Trigonometric Functions II (Draft August 007) Page of 7
4.0. Solving Quadratic Trigonometric Equations (Continued) Method 3: Identities and Factoring Solve for,0 + 0 3sin 9 0 Use an identity here:???? Method 4: Identities and Quadratic Formula Solve for,0 sec 5tan + Use an identity here: sec???? Advanced Functions: MHF4U -: Unit 4 Trigonometric Functions II (Draft August 007) Page 3 of 7
4.0. Solving Quadratic Trigonometric Equations (Solutions) In the following 4 eamples, you will deal with 4 different methods to deal with quadratic trigonometric equations. Method : Common Factor Solve for θ,0 θ tanθ θ tanθ tanθ θ tanθ 0 tan θ( θ ) 0 θ 0 θ or tanθ 0 θ ± θ n or θ n θ 0,, Method : Trinomial Factor Solve for,0 3sin sin 0 (3sin + )(sin ) 0 3sin + 0 or sin 0 sin 3 or sin 3.873 or 5.5535 or Advanced Functions: MHF4U -: Unit 4 Trigonometric Functions II (Draft August 007) Page 4 of 7
4.0. Solving Quadratic Trigonometric Equations (Solutions continued) Method 3: Identities and Factoring Solve for,0 0 3sin + 9 0 0( sin ) 3sin + 9 0 0sin 3sin 0 sin 0 or 5sin + 0 sin or sin 5 5, or 3.3430 or 6.088 6 6 Method 4: Identities and Quadratic Formula Solve for,0 sec + 5tan sec + 5 tan + 0 (+ tan ) + 5 tan + 0 tan + 5tan + 3 0 b± b 4ac From Quadratic Formula tan a tan 0.697 or tan 4.308.537 or 5.6743 or.799 or 4.9408 Advanced Functions: MHF4U -: Unit 4 Trigonometric Functions II (Draft August 007) Page 5 of 7
4.0.3 Solving Quadratic Trigonometric Equations: Practice Knowledge 0. Find the solutions on the interval [0, ]: ( a) 3sin 8sin 3 0 ( b) 5 + 6 8. Find the solutions on the interval [0, ]: ( a) cot ( b) sin + sin 0. Find solutions on the interval [0, ]: ( a) csc ( b) 4sin tan 3tan + 0sin 5 0 Hint: in part (b) one factor is tan + 5 3. Use an identity to find solutions on the interval [0, ]: ( a) sec tan 0 ( b) sec + tan 3 Application 4. Another model for a bouncing ball is ht () 4sin (8 t) where h is height measured in metres and t is time measured in seconds. When is the ball at a height of m? 5. Use a trigonometric identity to solve the following on the interval [0, ]:: sin Advanced Functions: MHF4U -: Unit 4 Trigonometric Functions II (Draft August 007) Page 6 of 7
4.0.3: Solving Quadratic Trigonometric Equations: Practice (Continued ) Communication 6. Compare and contrast the approach to solving a linear trigonometric equation and a quadratic trigonometric equation. 7. What is wrong with this solution? sin tan sin sin tan sin tan 5 or 4 4 Thinking 8. Find all solutions, if possible, of the below equation on the interval [0, ]: sin + 3 0 Answers: 3 7. ( a) 3.484,5.9433 ( b),,.588,5.3004 4 4 5 3. (a),,, (b) 0.83,.303 4 4 5 3 3. (a),,, (b) 0.848,.768,.935,4.9098 6 6 3 5 7 5 4. (a),,, (b), 4 4 4 4 4 4 5. 6. n t +, n 0,,,... 3 6 5 3,,, ; 7. Answers will vary 6 6 8. You divide one source of your roots. Sin() should be brought to LS and then factored 9. No solution Advanced Functions: MHF4U -: Unit 4 Trigonometric Functions II (Draft August 007) Page 7 of 7