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?  Do you think the review test we took on Friday improved your “learning” and grade for this grading period?

Chapter Review Test Results Monday 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 1 st period average: 70.7 to rd period average: 81.2 to th period average: 78 to 85  Weekly workshop research at UAH – 1 letter grade improvement in most cases

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

Weekly Plan  Monday – 1/27/14 Chapter Test Review – final thoughts  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

Learning Objectives for the Week!  UAH experience with precalculus courses!  Important Note:Students should not plan to operate heavy equipment this week!  Objectives: 1.Learn the proper way to do a mathematical proof – two line examples with explanations of “why” (versus what) 2.Learn how to use the fundamental trigonometric identities  Memorization will not required 3.Develop a “useful” strategy for proving identities  You will be allowed to reference this for quizzes/tests 4.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) 5.Gain confidence – reduce the overall fear of the word “proof” when doing mathematics! So, what is an Identity???

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 = x 2 + 2xy + y 2 0 = 0, = 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 180 0

Trigonometric Identities  Trigonometric identities are equalities that involve trigonometric functions and are true for every single value ( Geometrically – true for all angles in the unit circle) Pythagorean – sin 2 (x) + cos 2 (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

Most famous of all! Pythagorean Identity  sin 2 ( ) + cos 2 ( ) = 1

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)

#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

#83: cos(x + ) = cos(x)  Guess was 1.57 or

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

Fundamental Trig Identities  Quotient/Reciprocal  Pythagorean  Even-Odd = 1

For Homework/Review  Read Chapter 5.1  Page 586 to Page 593  Pay attention to examples!

Course of Study – ALEX Precalculus  33.) Prove the Pythagorean identity sin 2 (θ) + cos 2 (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]