Pre-calculus Chapter 5

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1 NAME: Pre-calculus Chapter 5 Date Day Lesson Assigned Due 3/25 Monday 5.1 & 5.2 Pg. 357:15-20, 30, 34, 37, 41, 49, 50, 51. Pg. 365: 3, 6, 9, 24-28, 36, 98, 100 3/27 Wednesday 5.3 Day 1 Pg. 376: Vocab 1-3, Ex: 9-24 mult of 3, /29 Friday 5.3 Day 2 4/9 Tuesday 5.4 & 5.5 4/11 Thursday Review Pg. 376: 33-46, 87, 90, 96, 97, 104, 108, 112, 116, 117 Pg. 384: 6, 16, 36, 40, 48, 72, 82, 104. Pg. 394: 2, 6, 14, 38, 57, 90, 102, 138, 146 Review Worksheet Extra Credit: pg. 400: 26, 36, even, 68, 76, 78, 92, 94, 110, 112 4/15 Monday Test Test 5.1 & 5.2 (one assignment) 5.3 Day Day & 5.5 (one assignment) Review Notes Extra Credit Notes: 1 PreCalculus CHAPTER 5

2 2 PreCalculus CHAPTER 5

3 5.1 Using Fundamental Identities In this chapter, you will learn how to use the fundamental identities to perform the following: Evaluate Trig Functions Simplify Trig Functions Develop additional Trig Identities The Fundamental Trigonometric Identities are provided for you on a separate sheet of paper. Refer to this sheet as we proceed to work on the following examples. Example 1: Show that Example 2: Simplify the expression using the fundamental identities Example 3: Factor the expression and use the fundamental identities to simplify. Example 4: Perform the multiplication and use the fundamental identities to simplify. ( ) 3 PreCalculus CHAPTER 5

4 5.2 Verifying Trig Identities In this section we will verify trig identities. In other words, prove that one side of the equation is equal to the other side (remember, do not cross the line!). Work on one side only. The key to verifying identities and solving equations is the ability to use the fundamental identities and the rules of algebra to rewrite trigonometric expressions. GUIDELINES FOR VERIFYING TRIG IDENTITIES 1. Work with one of the equation or the other side. It is often better to work with the most complicated side first. 2. Look for opportunities to factor an expression, get common denominators, foil, multiply by a conjugate, or create a denominator having only one term. 3. Look for opportunities to use the fundamental identities. Note which expressions are in the final expression you want. Sines and cosines pair up well, as do secants and tangents, and cosecants and cotangents. 4. If the preceding guidelines do not help, try converting all terms to sines and cosines. 5. Do not just sit and stare at the problem! Try something! Even paths that lead to dead ends can give you insights. Verify the following identities. Example 1: Example 2: Example 3: 4 PreCalculus CHAPTER 5

5 Example 4: ( ) ( ) 5 PreCalculus CHAPTER 5

6 5.3 Solving Trigonometric Equations In this section you will solve trigonometric equation. That is, you will solve for x. Since trig equations oscillate, there are an infinite number of solutions. You will learn how to write solutions when there are an infinite number of solutions and for some equations we will restrict the domain and write only those solutions that fit in that domain. Examples follow. Example 1: Solve, give the general solution. Example 2: Solve the following equation giving solutions only from [0, 2 ) Multiple Angles Find all solutions from [0, 2 ) Example 3a: Example 3b: 6 PreCalculus CHAPTER 5

7 Quadratic Type Example 4: Find all solutions from [0, 2 ) Factoring Example 5: Solve giving the general solutions. Rewriting with a Single Trig Function Example 6: Solve giving the general solutions. Confirm your answers graphically. 7 PreCalculus CHAPTER 5

8 Squaring and Converting to Quadratic Type Example 7: Find all solutions from [0, 2 ) 8 PreCalculus CHAPTER 5

9 5.4 Sum and Difference Formulas In this section, you will learn how to find exact values of all six trig functions when the angle measure is not one of the angles in your special triangles. You will do this by using the Sum and Difference Identities. Please refer to you Trigonometric Identity hand out. Example 1: Find the exact value of using your Sum and Difference Identities. Example 2: Find the exact value of Example 3: Find ( ) given that and (Hint: draw the triangles and use the Pythagorean Theorem to solve for the third side.) 9 PreCalculus CHAPTER 5

10 Example 4: Find all solutions from [0, 2 ) without a calculator. (hint: use the sum & difference identities for the cosine.) ( ) ( ) 1 0 PreCalculus CHAPTER 5

11 5.5 Multiple Angle Formulas In this section, you will learn to use the Double Angle and the Half-Angle identities to help find exact values of trigonometric functions. You will also use these multiple-angle identities to help you solve trig equations. Refer to your Trigonometric Identities handout for these formulas. We will also take a look at the Sum-to-Product Formula. Example 1: Find the exact values of (Hint: draw a triangle first and label all 3 sides). by using the double angle formulas. = = = 1 1 PreCalculus CHAPTER 5

12 Example 2: Find the exact values of a triangle and label all 3 sides). by using the half-angle formulas. (Hint: draw 1 2 PreCalculus CHAPTER 5

Syllabus ENGR 190 Introductory Calculus (QR)

Syllabus ENGR 190 Introductory Calculus (QR) Catalog Data: ENGR 190 Introductory Calculus (4 credit hours). Note: This course may not be used for credit toward the J.B. Speed School of Engineering B. S.