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YOUNGSTOWN CITY SCHOOLS MATH: PRECALCULUS UNIT 5: PROVING TRIGONOMETRIC IDENTITIES (2 WEEKS) 2013-2014 Synopsis: Students will be proving the Pythagorean identity and the addition and subtraction formulas for sine, cosine and tangent. After the proofs have been presented, they will use these identities to solve problems. STANDARDS F.TF.8. 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.TF.9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. MATH PRACTICES 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning LITERACY STANDARDS L-2 Communicate using correct mathematical terminology L-3 Read, discuss, and apply the mathematics found in literature including looking at the author’s purpose L-4 Listen to and critique peer explanations of reasoning MOTIVATION TEACHER NOTES 1. Read poem on trigonometry, attached on page 8 or on the link listed below. Students then write their own poems on trigonometry. http://wiki.answers.com/Q/Can_you_help_you_write_an_acrostic_poem_for_trigonometry 2. The broadcasting example found on page 437 of the teacher’s textbook, Glencoe Advanced Mathematical Concepts Precalculus with Applications is a good motivator. It explains interference in broadcasting and goes on to solve it in example 4 on page 440. 3. Preview expectations for the end of the Unit 4. Have student set both personal and academic goals for this Unit. TEACHING-LEARNING Vocabulary Sum identity for cosine Difference identity for cosine Pythagorean identity TEACHER NOTES Sum identify for sine Difference identity for sine Sum identity for tangent Difference identify for tangent 1. Students need to know the Law of Cosines before they can prove the sum and difference identities. This would also be a good time to present the Law of Sines since it seems to follow the material in Unit 6. So use section 5-6 and 5-7 of the textbook and only discuss the Law of Sines and its ambiguous case. Problems to look at are on page 316-317, numbers 11-18, 28, 31, 32, 33, 34; pages 324-325, numbers 11-37. The Law of Cosines is in section 5-8. Only discuss the Law of Cosines which would be examples 1 and 2. Problems to look at are on page 331-332, problems #11-18, 27, 31. 1 05/08/2013 YCS PRE-CALC UNIT 5 PROVING TRIG IDENTITIES 2013-14 TEACHING-LEARNING TEACHER NOTES 2. Referring to a unit circle, discuss with students the basic identity: and place them in small groups to arrive at a proof for this identity. Possibly give them the hint to use the basic definitions for . Present students with several problems using this identity. For example: given tan(x) = 12 and x is in the third quadrant, find sin(x) and cos(x). Ans: 3. Discuss the proof of the Pythagorean Identity, , using the diagram on the web site listed, if necessary: http://www.coolmath.com/lesson-pythagorean-identities-1.htm (F.TF.8, MP.1, MP.2, MP.3, MP.4, MP.7, L.2) 4. Have students derive the two other forms of this identity, stating reasons for their steps: and cos2(x) = 1 – sin2(x) (F.TF.8, MP.1, MP.2, MP.3, MP.4, MP.7, L,2, L,4) 5. After proving the three identities algebraically, take a graphical approach to the proofs. Examine the functions formed by taking each side of the equation and graphing them to show that they are identical. (F.TF.8, MP.1, MP.2, MP.4, MP.5, L,2) 6. Present to students the following problem or one similar to it: If cos x = and x is in the first quadrant, find the value for the sin x using the Pythagorean identity. Pose the question: What would happen if x is in the fourth quadrant? Why? Have students work several examples similar to the given example, finding values for sin x, cos x, and tan x given one trig ratio and the quadrant of the angle. Discuss what happens if the quadrant of the angle is not given. Worksheet #1 is included on page 9 for reinforcement. (F.TF.8, MP.1, MP.2, MP.4, MP.7, MP.8, L,2) 7. Engage students in the proof of the difference identity for cosines on page 437 of the textbook. An alternative proof of the sum and difference identities for cosine and sine are on the link: http://www.milefoot.com/math/trig/22anglesumidentities.htm . Listed below are several ways to engage students in proofs: List the steps of the proof, leaving some of the steps out for the students to fill in Write the identity that is to be proven and draw the picture on a piece of paper, hand students strips of paper with the steps and reasons written on them and have them put the steps in order underneath the identity and drawing Write up the proof with errors in it and have students correct the errors. (F.TF.9, MP.1, MP.2, MP.3, MP.4, MP.7, MP.8, L.2, L.4) 8. After students understand the proof of the difference identity for cosines, have them look at the proofs for the sum identity for cosine (page 438), sum identity for sine (page 439), difference identity for sine (page 439), sum identity for tangent and difference identity for tangent (problem #47 on page 444). (F.TF.9, MP.1, MP.2, MP.3, MP.4, MP.7, MP.8, L.2, L.4) 9. Students also need to be able to solve problems using these identities. The examples Attached on pages 10-12 and those in your textbook on page 442-444 may help to explain the procedure used to solve this type of problem. Work only those problems involving sine, cosine and tangent. (F.TF.9, MP.1, MP.2, MP.4, MP.7, MP.8, L.2) TRADITIONAL ASSESSMENT 1. Unit Tests: Multiple-Choice Questions TEACHER NOTES 2 05/08/2013 YCS PRE-CALC UNIT 5 PROVING TRIG IDENTITIES 2013-14 TEACHER CLASSROOM ASSESSMENT 1. Teacher Classroom Assessments 2. Smaller authentic assessments as you go along TEACHER NOTES TEACHER NOTES AUTHENTIC ASSESSMENT (30% of grade) 1. Have students evaluate goals for the unit. 2. Students are to find errors and correct them in problems 1 thru 4 and disprove an identity by giving a counterexample in problem 5 - - attached on page 4; Rubric and answers are on pages 5-7. (F.TF.8, F.TF.9, MP.1, MP.2, MP.3, MP.7, MP.8, L.2) 3 05/08/2013 YCS PRE-CALC UNIT 5 PROVING TRIG IDENTITIES 2013-14 AUTHENTIC ASSESSMENT PROBLEMS: 1. The Pythagorean identity states: Adding a cos2 (x) from each side we have By taking the square root of each side we have 2. Find the tan (x-y) given 0< x< 3. Simplify: cos (x and sin2 (x) = 1 – cos2 (x) sin (x) = 1 – cos (x) < y< π, sin x = ) cos (x )= cos x * cos + sin x * sin cos x * 1 + sin x * 0 = cos x = 4. Find the exact value for 4 05/08/2013 YCS PRE-CALC UNIT 5 PROVING TRIG IDENTITIES 2013-14 5. Prove: sin y * tan y = cos y is not a trig identity by using a counterexample AUTHENTIC ASSESSMENT RUBRIC ELEMENTS OF THE PROJECT Pythagorean identity 0 1 2 3 Did not attempt Corrected one error, no explanation Corrected both errors, no explanation Corrected both errors and gave explanation Difference identity for Did not tangent attempt Corrected one error, no explanation Corrected both errors, no explanation Corrected both errors and gave explanation Difference identity for Did not cosine attempt Corrected one error, no explanation Corrected both errors, no explanation Corrected both errors and gave explanation Sum identity for sine Did not attempt Corrected one error, no explanation Corrected both errors, no explanation Corrected both errors and gave explanation Disprove identity with counterexample Did not attempt Chose a value for y such that the identity was true Chose a value for y which disproved the identity, errors in the proof Chose a value for y which disproved the identity with no errors 5 05/08/2013 YCS PRE-CALC UNIT 5 PROVING TRIG IDENTITIES 2013-14 Answers for the Authentic Assessment: 1. The Pythagorean identity states: Adding a cos2 (x) from each side we have sin2 (x) = 1 – cos2 (x) By taking the square root of each side we have sin (x) = 1 – cos (x) 2 In the second line you are subtracting a cos (x) from each side. When you take the square root of each side you get sin (x) = root of each term. 2. Find the tan (x-y) given 0< x< and < y< π, sin x = Explanation: tan x = and the tan y = quadrant making it negative. 3. Simplify: cos (x , you cannot take the square since tangent is defined as and y is in the second ) cos (x )= cos x * cos + sin x * sin cos x * 0 + sin x * 1 = sin x When looking at the unit circle, cos = , sin 4. Find the exact value for 6 05/08/2013 YCS PRE-CALC UNIT 5 PROVING TRIG IDENTITIES 2013-14 When looking at a 45-45-90 triangle on the unit circle, sin 5. Prove: sin y * tan y = cos y is not a trig identity by using a counterexample. Students can choose a value for y, preferably one for which they know the sine, cosine and tangent and one that disproves the identity. Say you choose cos . Which means for y, then the identity becomes sin * tan = which is false so the identity does not work. 7 05/08/2013 YCS PRE-CALC UNIT 5 PROVING TRIG IDENTITIES 2013-14 TWO POEMS ON TRIGONOMETRY Tremendous labor is what it takes Rigorous worksheets when you start to accomplish Identities, not our own, but with six trigonometric functions Graphing sine cosine tangent along with their inverses Operations are still set with equalities and inequalities No other way but the mastery of the basic tools in circles Onward with complex, quadratics and its applications Meted with exponential and logarithmic functions Endeavor does not end because we'll meet Gauss-Jordan and Cramer To submerge with determinants, matrices - even augmented Ready or not the FUN will surely start Yes but with PATIENCE though bitter we'll surely PASS! [email protected] Time may give you pressure, Relax and let your mind do the work, In Trigo, it's all the complex equations, Give your best shot and solve them all, Over and over, repeat the solutions, Nothing should stop you, must concentrate, On a table with a paper and pencil, Must answer your best before time runs out, Erase your mistakes but learn from it, Try again, don't give up, Remember the lessons, reflect on them, You can do it, it's just a subject with numbers. ~Sarleen Castro Westfields International School, Philippines 8 05/08/2013 YCS PRE-CALC UNIT 5 PROVING TRIG IDENTITIES 2013-14 T/L #6: WORKSHEET #1 Using the Pythagorean identity, solve the following problems: 1. Sin θ = and 180< θ<270, find the cos θ and tan θ 2. Tan θ = and 180< θ<270, find the cos θ and sin θ 3. Cos θ = and 270< θ<360, find the sin θ and tan θ 4. Tan θ = and 5. Cos θ = , find the sin θ and cos θ , find sin θ and tan θ in terms of x 6. Sin θ = find cos θ and tan θ 7. Cos θ = , find sin θ and tan θ 8. Tan θ = , find sin θ and cos θ _____________________________________________________________________________________________ Answers: 1. cos θ = 2. cos θ = -½ and sin θ= 3. sin θ = and tan θ = 4. sin θ = 5. sin θ = and tan θ = 6. cos = ½ and tan 7. sin θ = 8. sin θ = 9 05/08/2013 YCS PRE-CALC UNIT 5 PROVING TRIG IDENTITIES 2013-14 T-L #9 Problems: Example 1 a. Show by producing a counterexample that cos (x - y) ≠ cos x - cos y. b. Show that the difference identity for cosine is true for the values used in part a. Answers: a. Let x = 0 and y = 4. First find cos (x - y) for x = 0 and y = 4. cos (x - y) = cos (0 - 4) Replace x with 0 and y with 4. = cos (-4) 0 - 4 = -4 2 = 2 Now find cos x - cos y. cos x - cos y = cos 0 - cos 4 Replace x with 0 and y with 4. 2 2- 2 = 1 - 2 or 2 So, cos (x - y) ≠ cos x - cos y. b. Show that cos (x - y) = cos x cos y + sin x sin y for x = 0 and y = 4. 2 First find cos (x - y). From part a, we know that cos (0 - 4) = 2 . Now find cos x cos y + sin x sin y. cos x cos y + sin x sin y = cos 0 cos 4 + sin 0 sin 4 = (1) Substitute for x and y. ( 22) + (0) ( 22) 2 = 2 Thus, the difference identity for cosine is true for x = 0 and y = 4. 10 05/08/2013 YCS PRE-CALC UNIT 5 PROVING TRIG IDENTITIES 2013-14 Example 2 Use the sum or difference identity for cosine to find the exact value of cos 825°. Answer: 825° = 2(360°) + 105° Symmetry identity, Case 1 cos 825° = cos 105° cos 105° = cos (60° + 45°) = cos 60° cos 45° - sin 60° sin 45° 60° and 45° are two common angles whose sum is 105°. Sum identity for cosine 1 2 3 2 =2 2 - 2 2 2- 6 4 = Therefore, cos 825° = 2- 6 4 . Example 3 12 15 Find the value of sin (x + y) if 0 < x < 2, 0 < y < 2, sin x = 13 and sin y = 17. Answer: In order to use the sum identity for sine, we need to know cos x and cos y. We can use a Pythagorean identity to determine the necessary values. sin2 α + cos2 α = 1 so cos2 α = 1 - sin2 α Pythagorean identity Since we are given that the angles in Quadrant I, the values of sine and cosine are positive. Therefore, cos α = 1 - sin2 α. cos x = = cos y = 1 12 13 2 25 5 169 or 13 1 15 17 2 64 8 or 289 17 Now substitute these values in to the difference identity for sine. sin (x + y) = sin x cos y + cos x sin y 12 8 5 15 = 13 17 + 13 17 = ( )( ) ( )( ) 171 = 221 or about 0.774 11 05/08/2013 YCS PRE-CALC UNIT 5 PROVING TRIG IDENTITIES 2013-14 Example 4 TRIGONOMETRY If the sine wave is shifted 2 units to the left, it coincides with the cosine wave. Therefore, cos = sin ( + 2 ) is an identity. Prove the identity. Answer: cos = sin ( + 2) cos = sin cos 2 + cos sin 2 cos = sin (0) + cos (1) cos = cos The identity is true. Example 5 Use the sum or difference identity for tangent to find the exact value of tan 345°. tan 345° = tan (300° + 45°) 300° and 45° are two common angles whose sum is 345°. tan 300° + tan 45° = 1 - tan 300° tan 45° Sum identity for tangent = tan 300° = - 3, tan 45° = 1 = Multiply by 1- 3 to simplify. 1- 3 12 05/08/2013 YCS PRE-CALC UNIT 5 PROVING TRIG IDENTITIES 2013-14