Concept Category 3 Trigonometric Functions

Transcription

1 Concept Category 3 Trigonometric Functions LT 3A I can prove the addition and subtraction formulas for sine, cosine, and tangent. I can use the addition and subtraction formulas for sine, cosine, and tangent to simplify trigonometric expressions, which may involve composite functions, and demonstrate equivalent relationships between trigonometric functions (trigonometric identities). I can prove the half angle and double angle formulas.

2 What question comes to mind?

3 What are your tools from Trig?

4 Given a point P(x,y) on the unit circle and x 2 +y 2 = 1. Verify that tan 2 θ + 1 = sec 2 θ Use your Trig knowledge from last Questions only try to come up year and 3 questions to create and with 3 execute a plan

5 Given a point P(x,y) on the unit circle and x 2 +y 2 = 1. Verify that tan 2 θ + 1 = sec 2 θ Point of confusion/hit a wall Write down questions you have Got your back Questions to approach questions didn t answered go Go back get to plan answered revise back to execution your plan Was there any relevant information you missed? Were there any assumptions or approximations that needed to be made? Did you contrast the current problem to a previous task/problem? How does the information in the approach inform your plan?

6 Unit Circle A. Definition of Unit Circle: Circle of unit radius, i.e., of radius 1. What are the Write all six six trig trig functions functions in in terms of x terms of sine and y and cosine?

7 B. Visual What are the key features of the Unit Circle?

8 III. Verifying Trigonometric Identities A. Identities How do these connect to the Unit Circle?

9 C. Process Verify the identity

10 C. Process Verify the identity

11 D. Purpose Solving Trig Equations 3cosθ + 3 = 2sin 2 θ Integrals in Calculus

12 Goal Problems What is the role of sine and cosine in proving trig identities? Verify the identity What strategies from CC1 manipulating expressions can be used to prove trig identities? Verify the identity What strategies from CC1 manipulating expressions can be used to prove trig identities? Verify the identity

13 Active Practice What is the role of sine and cosine in proving trig identities? P. 543 #48-64 What strategies from CC1 manipulating expressions can be used to prove trig identities? P. 543 #65-76 What strategies from CC1 manipulating expressions can be used to prove trig identities? P. 544 #77-82

14 What songs do you get stuck in your head?

15 A. More identities Sine cosine cosine sine Cosine cosine sign sine sine

16 How are these derived?

17 What is your action plan for this weekend? CC3 & CC4 Mock Final CC1 & CC2 Mock Final

18 Identities

19 How are these derived?

20 Active Practice Problems are grouped by similar problems, generally getting hard as you progress. Skip around as you work your way through #1-32. Goal: To be able to verify identities without notes or assistance. Trig identities should be memorized

21 Opening Timed Quiz (short quiz) Will begin 1 minute after bell. Take out something quickly to practice on: Post-it note/notebook/quarter sheet of paper Something to write with

22 Opening Timed Quiz (1 Minute) Identity Quiz Write Bread and Butter Equation Write Sum/Difference Equations If you have time, derive the other 2 Pythagorean identities. (Tan, Cot, Sec, Csc)

23 Identities

24 Goal Problems: Work Independantly. Start on a blank page with no identities written down. What is the role of sine and cosine in proving trig identities? Verify the identity What strategies from CC1 manipulating expressions can be used to prove trig identities? Verify the identity What strategies from CC1 manipulating expressions can be used to prove trig identities? Verify the identity

25 Goal Problems Pythagorean Identities: Try #9, 25 Odd/Even Identities: Try #37, 38 Reciprocal Identities (6 Trig Functions):Try #31, 32, 45, 44, Review #17-24 Using Sum & Difference & Double Angle/Proving Cofunction Identities Verify the identity Proving Identities If f(x) = sin x, show that Try #33, 39 Try #39

26 CoFunction Identities

27 Goal Problems: Work Independantly. Start on a blank page with no identities written down. What is the role of sine and cosine in proving trig identities? Verify the identity What strategies from CC1 manipulating expressions can be used to prove trig identities? Verify the identity What strategies from CC1 manipulating expressions can be used to prove trig identities? Verify the identity

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29 Active Practice What is the role of sine and cosine in proving trig identities? P. 551 #21-31 or P. 543 #48-64 What strategies from CC3 manipulating expressions can be used to prove trig identities? P. 552 #65 & 66 or P. 543 #65-76 What strategies from CC3 manipulating expressions can be used to prove trig identities? P. 544 #77-82 or P. 552 #55-58

30 Defense: Verifying Trig Identities 1. Definition 2. Visual & derivation (concept focus) 3. Process to verify a trig identity. (Procedure focus) 4. Purpose: why need to manipulate expressions(fluency) 5. Connections: Basic Trig Identities, Manipulating Expressions CC3 Reasoning in purple (use question prompts as a guide)

31 Concept Category 3 Trigonometric Functions LT 3B I can evaluate inverse trigonometric expressions. I can use inverse functions to solve trigonometric equations, with restricted and non-restricted domain, that arise in modeling contexts. I can evaluate the solutions using technology and interpret them in terms of the context. I can explain why restricting y = sinx or y = cosx to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

32 Problem

33 Problem What are your tools?

34 Problem

35 A. Defintion A. Inverse Trig Functions- For example with the sine function we would use the notation Why do we need inverse Trig functions?

36 A. Defintion A. Inverse Trig Functions- when trig functions are restricted to the domains below we can define the inverse functions. For example with the sine function we would use the notation What are the domain Why do we need and ranges for each inverse Trig functions? inverse function? Why do we need to restrict the domains?

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38 B. Visual

39 B. Visual

40 Examples Find all solutions of the equation 2 sinθ 1 = 0

41 Examples Solve the equation 5 sinθ cosθ + 4 cosθ = 0

42 C. Process

43 Solution to Goal Problem

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45

46 Goal Problems Quadratic Type/Factoring Solve Solving Trig Equations by using Identities Solve Applications with Trig equations In Philadelphia the number of hours of daylight on day t (where t is the number of days after January 1) is modeled by the function a. Which days of the year have about 10 h of daylight? b. How many days of the year have more than 10 h of daylight?

47 Active Practice What are the characteristics of a quadratic type function? What characteristics of the equation clue you in to whether or not the equation can be manipulated using trig identities? What is the role of domain restrictions in finding solutions to trig equations?

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49 Find the height of the mountain shown below.

50 Concept Category 3 Trigonometric Functions LT 3C I can prove the Law of Sines and Law of Cosines and use the Law of Sines and Law of Cosines to find unknown measurements in applications and evaluate the solutions in terms of the context. I can derive the formula A = 1/2 ab sinc for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side and find the area of any triangle using the formula.

51 What questions come to mind?

52 Task

53 A. Definition Law of Sines: In any triangle the length of the sides are proportional to the sines of the corresponding angles.

54 B. Visual Used for acute and oblique triangles when: AAS ASA

55 B. Visual

56 Example Find the height of the mountain shown below.

57 Task A wrecking ball crane has an arm length of 15 feet and has an angle of elevation of 40 degrees. If a swinging wrecking ball is extended on a wire of 12 feet, draw a diagram representing the angle of the wire as the wrecking ball hits the floor.

58 A. Definition Ambiguous Case of Law of Sines: When given two sides and the non-included angle of a triangle, there may be one triangle, two triangles or no triangles with the given properties.

59 C. Process

60 Example

61 Task

62 Goal Problems Using Law of Sines Find the length of AD

63 Goal Problems Solving Triangles Using Law of Sines Two wires tether a balloon to the ground, as shown. How high is the balloon above the ground?

64 Goal Problems Solving Triangles Using Law of Sines 554 ft.

65 Task

66 Error Analysis of Mock Final CC1 has a score. CC2 does not have a score yet. Take out a different colored pen. Swap tests with someone. You will need to correct their tests and help them analyze their errors. Using the answer Key, can you find their mistake? Is it a procedural or conceptual error? Label with P or C, any errors you find. Think about what they would need to do to correct the answer? Return the test to the original owner and compare solution methods. You will return the test back to Mr. Wong at the end of the period. Make a plan on your calendars what you need to start reviewing.

67 Task Suppose a boat leaves port, travels 10 miles, turns 20 degrees, and travels another 8 miles as shown in how far from port is the boat? (*Hint* Imagine if you are in the boat)

68 A. Definition - Using the Law of Cosine The Law of Sines cannot be used directly to solve triangles if we only know two sides and the angle between them, or if we only know three sides.

69 B. Visual

70 A. Definition - Using the Law of Cosine The Law of Sines cannot be used directly to solve triangles if we only know two sides and the angle between them, or if we only know three sides.

71 C. Process Solve the triangle given: a = 51, b = 25, c = 29

72 Task Suppose a boat leaves port, travels 10 miles, turns 20 degrees, and travels another 8 miles as shown in how far from port is the boat? (*Hint* Imagine if you are in the boat)

73 B. Visual

74 Goal Problems Solving Triangles Using Law of Cosine Solving Triangles Using Law of Cosine 1.) C = 121, a = 21, b = 37 2.) a = 3.1, b = 3.5, c = 5

75 A. Definition Bearing: An acute angle that is measured from due north or due south.

76 B. Visual