Turn-In Assignment Instructions for Trig Identities
Objective: To develop sufficient skills so that you can establish or reject whether a given trigonometric equation is an identity or not, using the textbook-listed identities in Sections 6.1 (page 606) and 6.3 (page 634).
Due: End of Module 5. Deliverable: A Nicely handwritten, scanned PDF file or a Word document using the Math Equation Writer built-into Word, Or MathType, or Google Docs using its Math Equation writer, or a Mac equivalent.
Point Value: This assignment will be worth the same as a MyMathLab Module HW.
Expected Time toComplete this Learning Module: 5 hours
So, why create the Trig Identity Learning Module and not use the MyMathLab problems available for Sections 6.1 – 6.3? MyMathLab really can’t ask the right questions, since verifying a trig identity is more of an essay than finding a specific solution. This happens to be ideal, though, for turn-in exercises. In fact, most of the learning module directs you to specific parts of the textbook to answer some questions on the reading and accomplish some basic verifications. Then, you’ll accomplish “homework problems” chosen from the textbook exercises.
Part 1: Understanding why Identities in Trigonometry are important.
Task 1: Read pages 606 – 614, hopping over the examples at this point; answer the following questions, and cite the page you found your answer:
1. What does the text say is a key purpose of using trig identities?
2. In your own words, what does the book say about solving conditional equations vs verifying identities?
3. When attempting to verify that a given equation is an identity, am I allowed to multiply or add the same quantity to both sides of the equation? Explain your reason(s).
Part 2: Identifying the most useful techniques when verifying trig identities
1. Now, review the eight examples in Section 6.1 and write down the various techniques Blitzer uses to verify that the given trig equations are identities.
2. Of the techniques you identified, which did Blitzer say was the one or two most often used? In which examples did Blitzer make use of that or those technique(s)?
3. Did any of the Fundamental Trig identities in this section define any way to change one angle into a different one?
Now, turn to page 634 (Chapter 6, Section 3) and the table of Principal Trig Identities. What is “principally” being manipulated in this set of formulas?
It is clear that the Fundamental Set of identities in Section 6.1 combined with the Principal Set of identities in Section 6.3 have a lot of math power to find EQUIVALENT Trig expressions (remember that in week one, we pointed out that Equivalent Algebraic Expressions produced equations that were Identities. We can also say the reverse: that Identities identify equivalent algebraic or trig expressions that can be substituted one for the other to solve a conditional equation.
Well, that’s what Section 6.5 is all about—Solving trig equations. SO why do we skip Section 6.4? Time, and yet another set of identities would be a bit of overkill; they can be just as useful as the others, though. And THAT’S the Big Picture for this chapter.
Part 3: Verifying some trig identities to gain familiarity with the techniques in Part 2.
These problems are to be turned in. As you work each problem, identify which of the eight examples in the section is most helpful in solving the problem, AND ALSO WHICH FUNDAMENTAL IDENTITIES YOU ARE USING TO VERIFY EACH IDENTITY in the exercises.
Section 6.1, Problems 3, 7, 11, 16, 21, 26, 31, 36, 41.
Part 4: The Most Important Trig Identity in the Universe.
From the two special triangles (the isosceles right and the 30-60-90 triangle), we know EXACT trig values for all angles that are multiples of ?/6 and ?/4 (all multiples of 30 degrees and 45 degrees, respectively). Can we find EXACT trig values for all angles that are multiples of ?/12 (15 degrees)? With the Most Important Trig Identity in the Universe, we can. In fact, all the sum and difference of angles identities, all the double angle identities and all the half-angle identities and all the Power Reduction identities begin with this Most Important Identity in the Universe.
Blitzer takes two pages to verify it.Let’s turn to page 617, the section titled “The Cosine of the Difference of Two Angles”. Much of matrix theory and vector arithmetic also depend on this formula, which is an identity.
1. First, look at Figure 1a and 1b. Describe in words what Blitzer did between 6.1a and 6.1b.
2. Why is the Unit Circle, as opposed to a larger or smaller circle centered elsewhere, preferred?
3. On page 654, Blitzer has TWO separate formulas for the length of PQ. How did he build them?
4. What Fundamental Identity plays a key role in simplifying each formula for the length of PQ?
5. What are the final steps Blitzer takes to derive the Most Important Trig Identity in the Universe?
6. State that identity here. Use a large font and Bold weight to write it.
Part 5:Practice using it and the related sum/difference identities for sine and tangent. Same rules apply as for Part 3.
Exercise 6.2, Problems 2, 3, 9, 15, 24, 30, 33, 36, 39 (pp. 623-624). The last three are VERY IMPORTANT identities.Add in three problems from 57 – 63of your choosing(p. 624), but find only “Part a” for one of them, “Part b” for the second of them, and “Part c” for the third of them. QUADRANT INFO IS VERY IMPORTANT HERE.
That completes the assignment. Upload your electronic file containing the answers to all the Concept Questions and the Exercises from Parts 3 and 5 (some, but not many, of the Section 6.3 exercises also happen to be problems in the MyMathLab Module 5 HW. Hey, you get two shots at those problems.) If for some reason, your upload fails, send it as an attachment via email to your instructor. YOUR FILE NAME SHOULD BE LASTNAME LEARNING MODULE.extension like .docx, .doc, .pdf. MAC USERS: .doc or pdf, please. Your instructor may or may not have a Mac.
