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?

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

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 xx 11 2

xx 1)1( 2

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

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 = 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

Trigonometric Identitieshttp://www.purplemath.com/modules/idents.htm

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 – 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

Most famous of all!Pythagorean Identity

sin2( ) + cos2( ) = 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

2

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 – 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]