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Warm-Up Activity Write yourself a quick note!

Warm-Up Activity Write yourself a quick note!. Did you enjoy working problems on your desktop last week? Did the group work we did last week on Chapter 4 material help you better understand the concepts?

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Warm-Up Activity Write yourself a quick note!

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  1. Warm-Up ActivityWrite yourself a quick note! • Did you enjoy working problems on your desktop last week? • Did the group work we did last week on Chapter 4 material help you better understand the concepts? • Do you think the review test we took on Friday improved your “learning” and grade for this grading period?

  2. Chapter Review Test ResultsMonday 1/27/14 • Goal of review last week – Think and Learn vs. just doing the work! • Learn best with interaction! • Improvement in all 3 classes • 1st period average: 70.7 to 79.2 • 3rd period average: 81.2 to 87.8 • 5th period average: 78 to 85 • Weekly workshop research at UAH – 1 letter grade improvement in most cases

  3. Final Thoughts on Chapter 4 • On a test, read the directions! • Show your work = extra points! • Visualize - draw lots of pictures! • Content clarifications: • Reference angles are always positive, and there are infinitely many! • Learn to work with radians – it is actually easier than degrees! • Bearings – angles are to N/S axis in this course! • sin/cos graphs/key points – common denominators! • Creative math examples – interesting but not very useful or correct • (1)2 = 1, not 2

  4. Weekly Plan • Monday – 1/27/14 • Chapter Test Review – final thoughts • http://www.youtube.com/watch?v=ZS6YAViGft0 • Introduction to Identities – Learning objectives • What is an identity? • What are the fundamental trigonometric identities? • Tuesday – 1/28/14 • Develop a useful strategy for proving identities • Work examples – “I do”, “We do” • Wednesday Group Work • “Y’all Do” - Work trig puzzles/make group presentations • Thursday PreCal Workshop – 7 am to 8 am • Friday – 1/24/14 • Quiz on Section 5.1 – prove a couple of identities • Move on to Section 5.2 – Apply Sum/Difference Identities

  5. Learning Objectives for the Week! • UAH experience with precalculus courses! • Important Note:Students should not plan to operate heavy equipment this week! • Objectives: • Learn the proper way to do a mathematical proof – two line examples with explanations of “why” (versus what) • Learn how to use the fundamental trigonometric identities • Memorization will not required • Develop a “useful” strategy for proving identities • You will be allowed to reference this for quizzes/tests • Experience the personal satisfaction of proving an identity • Expect to make mistakes , and no two proofs may look exactly the same (see page AA51 in book) • Gain confidence – reduce the overall fear of the word “proof” when doing mathematics! So, what is an Identity???

  6. What is an identity? • Tautology – from greek logic – defined as a formula which is true in every possible interpretation. • A mathematical identity is defined as an expression that is always true for all possible values of x and y • (x+y)2 = x2 + 2xy + y2 • 0 = 0, 2 + 3 = 5 (in decimal) • An equation can be true for specific values of x, but not for every value • 3x = 12 if and only if x = 4 • cos(x) = -1 if and only if x = or 1800

  7. Trigonometric Identitieshttp://www.purplemath.com/modules/idents.htm • Trigonometric identitiesare equalities that involve trigonometric functions and are true for every single value (Geometrically – true for all angles in the unit circle) • Pythagorean – sin2(x) + cos2(x) = 1 • Identities are useful in simplifying algebraic expressions – the two sides are interchangeable at any time • These will be useful in section 5.5 when we solve trigonometric equations • In calculus, an important application involves integration of functions – trigonometric functions can be substituted and simplified using identities

  8. Most famous of all!Pythagorean Identity • sin2( ) + cos2( ) = 1

  9. Think/Pair/Share • Page 595 – Problem # 80 and # 83 • Work with your neighbor – use a graphing calculator to graph each side of the equation – radian mode, zoom 7 (Ztrig), discuss the difference between the two.. #80: y1 = sin(x) y2 = -cos(x)tan(-x) #83: y1 = cos(x + ) y2 = cos(x)

  10. #80: sin(x) = -cos(x)tan(-x) • LHS = sin(x), RHS = -cos(x)tan(-x) • Strategy #1: start with most complicated side first • Strategy #2: look for useful identities

  11. #83: cos(x + ) = cos(x) • Guess was 1.57 or

  12. Proving (Establish) Identities • Terminology • LHS = Left Hand Side • RHS = Right Hand Side • LHS = RHS proves the identity • Three approaches • Work LHS – make it look like RHS • Work RHS – make it look like LHS • Work Both, then show LHS = RHS

  13. Fundamental Trig Identities = 1 • Quotient/Reciprocal • Pythagorean • Even-Odd

  14. For Homework/Review • Read Chapter 5.1 • Page 586 to Page 593 • Pay attention to examples!

  15. Course of Study – ALEXPrecalculus • 33.) Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1, and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle. [F-TF8] (Alabama) • 27.) Use the sum, difference, and half-angle identities to find the exact value of a trigonometric function. (Alabama) • 34.) (+) Prove the addition and subtraction formulas for sine, cosine, and tangent, and use them to solve problems. [F-TF9]

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