Lesson 17 NYS COMMON CORE MATHEMATICS CURRICULUM M2 ALGEBRA II Lesson 17: Trigonometric Identity Proofs Student Outcomes ๏ง Students see derivations and proofs of the addition and subtraction formulas for sine and cosine. ๏ง Students prove some simple trigonometric identities. Lesson Notes The lesson starts with students looking for patterns in a table to make conjectures about the formulas for sin(๐ผ + ๐ฝ) and cos(๐ผ + ๐ฝ). From these formulas, students can quickly deduce the formulas for sin(๐ผ โ ๐ฝ) and cos(๐ผ โ ๐ฝ). The teacher gives proofs of important formulas, and then students prove some simple trigonometric identities. The lesson highlights MP.3 and MP.8, as students look for patterns in repeated calculations and construct arguments about the patterns they find. Scaffolding: ๏ง Students should have access to calculators. ๏ง The teacher may want to model a few calculations at the beginning. ๏ง Pairs of students who fill in the table quickly should be encouraged to help those who might be struggling. Classwork Opening Exercise (10 minutes) ๏ง Once a few students have filled in the table, one or two might be encouraged to share their entries with the class because when all students have the entries, they are in a better position to discover the rule. Students should work in pairs to fill out the table and look for patterns. They MP.8 should be looking for columns whose entries might be combined to yield the entries in the column for sin(๐ผ + ๐ฝ). Opening Exercise We have seen that ๐ฌ๐ข๐ง(๐ถ + ๐ท) โ ๐ฌ๐ข๐ง(๐ถ) + ๐ฌ๐ข๐ง(๐ท). So, what is ๐ฌ๐ข๐ง(๐ถ + ๐ท)? Begin by completing the following table: ๐ถ ๐ท ๐ฌ๐ข๐ง(๐ถ) ๐ฌ๐ข๐ง(๐ท) ๐ฌ๐ข๐ง(๐ถ + ๐ท) ๐ฌ๐ข๐ง(๐ถ) ๐๐จ๐ฌ(๐ท) ๐ฌ๐ข๐ง(๐ถ) ๐ฌ๐ข๐ง(๐ท) ๐๐จ๐ฌ(๐ถ) ๐๐จ๐ฌ(๐ท) ๐๐จ๐ฌ(๐ถ) ๐ฌ๐ข๐ง(๐ท) ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ โ๐ ๐ โ๐ ๐ ๐ ๐ ๐ ๐ โ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ โ๐ ๐ ๐ ๐ ๐ โ๐ ๐ โ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ โ๐ ๐ ๐ ๐ โ๐ + โ๐ ๐ โ๐ ๐ โ๐ ๐ โ๐ ๐ โ๐ ๐ ๐ ๐ ๐ ๐ โ๐ ๐ โ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ โ๐ ๐ โ๐ ๐ โ๐ ๐ โ๐ ๐ ๐ ๐ ๐ ๐ โ๐ ๐ ๐ ๐ ๐ ๐ โ๐ ๐ โ๐ ๐ โ๐ + โ๐ ๐ โ๐ ๐ โ๐ ๐ โ๐ ๐ โ๐ ๐ Lesson 17: Trigonometric Identity Proofs This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG II-M2-TE-1.3.0-08.2015 293 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 17 NYS COMMON CORE MATHEMATICS CURRICULUM M2 ALGEBRA II MP.8 Ask students to write an equation that describes how the entries in other columns might be combined to yield the entries in the shaded column. Scaffolding: ๏ง If no student offers the identity, the teacher might suggest that students look for two columns whose entries sum to yield sin(๐ผ + ๐ฝ). ๏ง It might even be necessary for the teacher to point at the two columns. The identity they are looking for in the table is the following: sin(๐ผ + ๐ฝ) = cos(๐ผ) sin(๐ฝ) + sin(๐ผ) cos(๐ฝ). Emphasize in the discussion that the proposed identity has not been proven; it has only been tested for some specific values of ๐ผ and ๐ฝ. Its status now is as a conjecture. The conjecture is strengthened by the following observation: Because ๐ผ and ๐ฝ play the same role in (๐ผ + ๐ฝ), they should not play different roles in any formula for the sine of that sum. In the conjecture, if ๐ผ and ๐ฝ are interchanged, the formula remains essentially the same. That symmetry helps make the conjecture more plausible. Use the following table to formulate a conjecture for ๐๐จ๐ฌ(๐ถ + ๐ท): MP.8 ๐ถ ๐ท ๐๐จ๐ฌ(๐ถ) ๐๐จ๐ฌ(๐ท) ๐๐จ๐ฌ(๐ถ + ๐ท) ๐ฌ๐ข๐ง(๐ถ) ๐๐จ๐ฌ(๐ท) ๐ฌ๐ข๐ง(๐ถ) ๐ฌ๐ข๐ง(๐ท) ๐๐จ๐ฌ(๐ถ) ๐๐จ๐ฌ(๐ท) ๐๐จ๐ฌ(๐ถ) ๐ฌ๐ข๐ง(๐ท) ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ โ๐ ๐ ๐ ๐ ๐ ๐ โ๐ ๐ ๐ ๐ ๐ ๐ โ๐ ๐ ๐ ๐ ๐ ๐ ๐ โ๐ ๐ โ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ โ๐ ๐ โ๐ ๐ โ๐ โ โ๐ ๐ โ๐ ๐ โ๐ ๐ โ๐ ๐ โ๐ ๐ ๐ ๐ ๐ ๐ โ๐ ๐ โ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ โ๐ ๐ ๐ ๐ ๐ ๐ โ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ โ๐ ๐ โ๐ ๐ โ๐ ๐ โ๐ ๐ โ๐ ๐ โ ๐ ๐ โ๐ โ โ๐ ๐ Again, ask students to write an equation that describes how the entries in other columns might be combined to yield the entries in the shaded column. The identity they are looking for in the table is the following: cos(๐ผ + ๐ฝ) = cos(๐ผ) cos(๐ฝ) โ sin(๐ผ) sin(๐ฝ). Again, in a class discussion of the exercise after students have looked for a pattern, if no student comes up with that identity, the teacher may want to point at the two columns whose entries differ to yield cos(๐ผ + ๐ฝ). It should be repeated that the proposed identity has not been proven; it has only been tested for some specific values of ๐ผ and ๐ฝ. It is a conjecture. This conjecture, too, is strengthened by the observation that because ๐ผ and ๐ฝ play the same role in (๐ผ + ๐ฝ), they should not play different roles in any formula for the cosine of that sum. And again, if ๐ผ and ๐ฝ are interchanged in the conjectured formula, it remains essentially the same. That symmetry helps make the conjecture more plausible. Lesson 17: Trigonometric Identity Proofs This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG II-M2-TE-1.3.0-08.2015 294 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 17 NYS COMMON CORE MATHEMATICS CURRICULUM M2 ALGEBRA II Examples 1โ2 (15 minutes): Formulas for sin(๐ผ + ๐ฝ) and cos(๐ผ + ๐ฝ) Scaffolding: Examples 1โ2: Formulas for ๐ฌ๐ข๐ง(๐ถ + ๐ท) and ๐๐จ๐ฌ(๐ถ + ๐ท) 1. One conjecture is that the formula for the sine of the sum of two numbers is ๐ฌ๐ข๐ง(๐ถ + ๐ท) = ๐ฌ๐ข๐ง(๐ถ) ๐๐จ๐ฌ(๐ท) + ๐๐จ๐ฌ(๐ถ) ๐ฌ๐ข๐ง(๐ท). The proof can be a little long, but it is fairly straightforward. We will prove only the case when the two numbers are positive, and their ๐ sum is less than . ๐ a. Let ๐ถ and ๐ท be positive real numbers such that ๐ < ๐ถ + ๐ท < b. Construct rectangle ๐ด๐ต๐ถ๐ท such that ๐ท๐น = ๐, ๐โ ๐ท๐ธ๐น = ๐๐°, ๐โ ๐น๐ท๐ธ = ๐ท, and ๐โ ๐ธ๐ท๐ด = ๐ถ. See the figure on the right. c. Fill in the blanks in terms of ๐ถ and ๐ท: i. ๐โ ๐น๐ท๐ถ = . ๐ . ๐ Because the proofs of the addition and subtraction formulas for sine and cosine can be complicated, only the proof of the sine addition formula is presented in detail here. Advanced students might be asked to prove, in much the same fashion, any of the other three formulas, as well as by deriving them from the sine addition formula. Problem 1 in the Problem Set concerns one of those proofs. ๐ โ๐ถโ๐ท ๐ ii. ๐โ ๐ท๐น๐ถ = . ๐ถ+๐ท iii. Therefore, ๐ฌ๐ข๐ง(๐ถ + ๐ท) = ๐ท๐ถ. iv. ๐น๐ธ = ๐ฌ๐ข๐ง(______). ๐ท v. ๐ท๐ธ = ๐๐จ๐ฌ(______). ๐ท d. Letโs label the angle and length measurements as shown. e. Use this new figure to fill in the blanks in terms of ๐ถ and ๐ท: i. Why does ๐ฌ๐ข๐ง(๐ถ) = ๐ด๐ธ ? ๐๐จ๐ฌ(๐ท) The length of the hypotenuse of โณ ๐ท๐ธ๐ด is ๐๐จ๐ฌ(๐ท), and ๐ด๐ธ is the length of the side opposite ๐ถ. ii. Therefore, ๐ด๐ธ = . ๐ฌ๐ข๐ง(๐ถ)๐๐จ๐ฌ(๐ท) iii. ๐โ ๐น๐ธ๐ต = . ๐ถ Lesson 17: Trigonometric Identity Proofs This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG II-M2-TE-1.3.0-08.2015 295 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 17 NYS COMMON CORE MATHEMATICS CURRICULUM M2 ALGEBRA II f. Now, consider โณ ๐น๐ธ๐ต. Since ๐๐จ๐ฌ(๐ถ) = ๐ธ๐ต = i. ๐ธ๐ต , ๐ฌ๐ข๐ง(๐ท) . ๐๐จ๐ฌ(๐ถ)๐ฌ๐ข๐ง(๐ท) 2. a. Label these lengths and angle measurements in the figure. b. Since ๐ด๐ต๐ถ๐ท is a rectangle, ๐ถ๐ท = ๐ด๐ธ + ๐ธ๐ต. c. Thus, ๐ฌ๐ข๐ง(๐ถ + ๐ท) = ๐ฌ๐ข๐ง(๐ถ) ๐๐จ๐ฌ(๐ท) + ๐๐จ๐ฌ(๐ถ) ๐ฌ๐ข๐ง(๐ท). Note that we have only proven the formula for the sine of the sum of two real numbers ๐ถ and ๐ท in the case where ๐ ๐ ๐ < ๐ถ + ๐ท < . A proof for all real numbers ๐ถ and ๐ท breaks down into cases that are proven similarly to the case we have just seen. Although we are omitting the full proof, this formula holds for all real numbers ๐ถ and ๐ท. Scaffolding: For any real numbers ๐ถ and ๐ท, ๐ฌ๐ข๐ง(๐ถ + ๐ท) = ๐ฌ๐ข๐ง(๐ถ)๐๐จ๐ฌ(๐ท) + ๐๐จ๐ฌ(๐ถ)๐ฌ๐ข๐ง(๐ท). 3. Now, letโs prove our other conjecture, which is that the formula for the cosine of the sum of two numbers is A wall poster with all four sum and difference formulas will help students keep these formulas straight. ๐๐จ๐ฌ(๐ถ + ๐ท) = ๐๐จ๐ฌ(๐ถ) ๐๐จ๐ฌ(๐ท) โ ๐ฌ๐ข๐ง(๐ถ) ๐ฌ๐ข๐ง(๐ท). ๐ Again, we will prove only the case when the two numbers are positive, and their sum is less than . This time, we ๐ will use the sine addition formula and identities from previous lessons instead of working through a geometric proof. Fill in the blanks in terms of ๐ถ and ๐ท: Let ๐ถ and ๐ท be any real numbers. Then, ๐ ๐๐จ๐ฌ(๐ถ + ๐ท) = ๐ฌ๐ข๐ง ( โ (__________)) ๐ = ๐ฌ๐ข๐ง((__________) โ ๐ท) = ๐ฌ๐ข๐ง((__________) + (โ๐ท)) = ๐ฌ๐ข๐ง(__________) ๐๐จ๐ฌ(โ๐ท) + ๐๐จ๐ฌ(__________) ๐ฌ๐ข๐ง(โ๐ท) = ๐๐จ๐ฌ(๐ถ) ๐๐จ๐ฌ(โ๐ท) + ๐ฌ๐ข๐ง(๐ถ) ๐ฌ๐ข๐ง(โ๐ท) = ๐๐จ๐ฌ(๐ถ) ๐๐จ๐ฌ(๐ท) โ ๐ฌ๐ข๐ง(๐ถ) ๐ฌ๐ข๐ง(๐ท). The completed proof should look like the following: ๐ ๐๐จ๐ฌ(๐ถ + ๐ท) = ๐ฌ๐ข๐ง ( โ (๐ถ + ๐ท)) ๐ ๐ = ๐ฌ๐ข๐ง (( โ ๐ถ) โ ๐ท) ๐ ๐ = ๐ฌ๐ข๐ง (( โ ๐ถ) + (โ๐ท)) ๐ ๐ ๐ = ๐ฌ๐ข๐ง ( โ ๐ถ) ๐๐จ๐ฌ(โ๐ท) + ๐๐จ๐ฌ ( โ ๐ถ) ๐ฌ๐ข๐ง(โ๐ท) ๐ ๐ = ๐๐จ๐ฌ(๐ถ) ๐๐จ๐ฌ(โ๐ท) + ๐ฌ๐ข๐ง(๐ถ) ๐ฌ๐ข๐ง(โ๐ท) = ๐๐จ๐ฌ(๐ถ) ๐๐จ๐ฌ(๐ท) โ ๐ฌ๐ข๐ง(๐ถ) ๐ฌ๐ข๐ง(๐ท). Lesson 17: Trigonometric Identity Proofs This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG II-M2-TE-1.3.0-08.2015 296 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 17 NYS COMMON CORE MATHEMATICS CURRICULUM M2 ALGEBRA II For all real numbers ๐ถ and ๐ท, ๐๐จ๐ฌ(๐ถ + ๐ท) = ๐๐จ๐ฌ(๐ถ)๐๐จ๐ฌ(๐ท) โ ๐ฌ๐ข๐ง(๐ถ)๐ฌ๐ข๐ง(๐ท). Exercises 1โ2 (6 minutes): Formulas for ๐ฌ๐ข๐ง(๐ถ โ ๐ท) and ๐๐จ๐ฌ(๐ถ โ ๐ท) In these exercises, formulas for the sine and cosine of the difference of two angles are developed from the formulas for the sine and cosine of the sum of two angles. Exercises 1โ2: Formulas for ๐ฌ๐ข๐ง(๐ถ โ ๐ท) and ๐๐จ๐ฌ(๐ถ โ ๐ท) 1. Rewrite the expression ๐ฌ๐ข๐ง(๐ถ โ ๐ท) as ๐ฌ๐ข๐ง(๐ถ + (โ๐ท)). Use the rewritten form to find a formula for the sine of the difference of two angles, recalling that the sine is an odd function. Scaffolding: ๏ง To help students understand the difference formulas, consider giving them some examples to calculate. ๏ง sin ( ๐ ๐ ๐ ๐ = sin ( ) cos ( ) โ cos ( ) sin ( ) 4 6 4 6 Let ๐ถ and ๐ท be any real numbers. Then, ๐ฌ๐ข๐ง(๐ถ + (โ๐ท)) = ๐ฌ๐ข๐ง(๐ถ) ๐๐จ๐ฌ(โ๐ท) + ๐๐จ๐ฌ(๐ถ) ๐ฌ๐ข๐ง(โ๐ท) = ๐ฌ๐ข๐ง(๐ถ) ๐๐จ๐ฌ(๐ท) โ ๐๐จ๐ฌ(๐ถ) ๐ฌ๐ข๐ง(๐ท). Therefore, ๐ฌ๐ข๐ง(๐ถ โ ๐ท) = ๐ฌ๐ข๐ง(๐ถ) ๐๐จ๐ฌ(๐ท) โ ๐๐จ๐ฌ(๐ถ) ๐ฌ๐ข๐ง(๐ท) for all real numbers ๐ถ and ๐ท. 2. Now, use the same idea to find a formula for the cosine of the difference of two angles. Recall that the cosine is an even function. Let ๐ถ and ๐ท be any real numbers. Then, ๐ ๐ ๐ ) = sin ( โ ) 12 4 6 = 1 1 1 โ3 โ โ 2 โ2 โ2 2 โ โ2โ3 โ2 โ 4 4 โ2(โ3 โ 1) = 4 = ๏ง cos ( ๐ ๐ ๐ ) = cos ( โ ) 12 4 6 ๐ ๐ ๐ ๐ = cos ( ) cos ( ) + sin ( ) sin ( ) 4 6 4 6 ๐๐จ๐ฌ(๐ถ โ ๐ท) = ๐๐จ๐ฌ(๐ถ + (โ๐ท)) = ๐๐จ๐ฌ(๐ถ) ๐๐จ๐ฌ(โ๐ท) โ ๐ฌ๐ข๐ง(๐ถ) ๐ฌ๐ข๐ง(โ๐ท) = ๐๐จ๐ฌ(๐ถ) ๐๐จ๐ฌ(๐ท) + ๐ฌ๐ข๐ง(๐ถ) ๐ฌ๐ข๐ง(๐ท). Therefore, ๐๐จ๐ฌ(๐ถ โ ๐ท) = ๐๐จ๐ฌ(๐ถ) ๐๐จ๐ฌ(๐ท) + ๐ฌ๐ข๐ง(๐ถ) ๐ฌ๐ข๐ง(๐ท) for all real numbers ๐ถ and ๐ท. For all real numbers ๐ถ and ๐ท, = 1 1 1 โ3 + โ โ2 2 โ2 2 โ โ2โ3 โ2 + 4 4 โ2(โ3 + 1) = 4 = ๐ฌ๐ข๐ง(๐ถ โ ๐ท) = ๐ฌ๐ข๐ง(๐ถ)๐๐จ๐ฌ(๐ท) โ ๐๐จ๐ฌ(๐ถ)๐ฌ๐ข๐ง(๐ท), and ๐๐จ๐ฌ(๐ถ โ ๐ท) = ๐๐จ๐ฌ(๐ถ)๐๐จ๐ฌ(๐ท) + ๐ฌ๐ข๐ง(๐ถ)๐ฌ๐ข๐ง(๐ท). Lesson 17: Trigonometric Identity Proofs This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG II-M2-TE-1.3.0-08.2015 297 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 17 M2 ALGEBRA II Exercises 3โ5 (10 minutes) These exercises make use of the formulas proved in the examples. Students should work on these exercises in pairs. Use the sum and difference formulas to do the following: Exercises 3โ5 3. Derive a formula for ๐ญ๐๐ง(๐ถ + ๐ท) in terms of ๐ญ๐๐ง(๐ถ) and ๐ญ๐๐ง(๐ท), where all of the expressions are defined. Hint: Use the addition formulas for sine and cosine. Let ๐ถ and ๐ท be any real numbers so that ๐๐จ๐ฌ(๐ถ) โ ๐, ๐๐จ๐ฌ(๐ท) โ ๐, and ๐๐จ๐ฌ(๐ถ + ๐ท) โ ๐. By the definition of tangent, ๐ญ๐๐ง(๐ถ + ๐ท) = ๐ฌ๐ข๐ง(๐ถ+๐ท) . ๐๐จ๐ฌ(๐ถ+๐ท) Using sum formulas for sine and cosine, we have ๐ฌ๐ข๐ง(๐ถ + ๐ท) ๐ฌ๐ข๐ง(๐ถ) ๐๐จ๐ฌ(๐ท) + ๐๐จ๐ฌ(๐ถ) ๐ฌ๐ข๐ง(๐ท) = . ๐๐จ๐ฌ(๐ถ + ๐ท) ๐๐จ๐ฌ(๐ถ) ๐๐จ๐ฌ(๐ท) โ ๐ฌ๐ข๐ง(๐ถ) ๐ฌ๐ข๐ง(๐ท) Dividing numerator and denominator by ๐๐จ๐ฌ(๐ถ) ๐๐จ๐ฌ(๐ท) gives ๐ฌ๐ข๐ง(๐ถ) ๐๐จ๐ฌ(๐ท) + ๐๐จ๐ฌ(๐ถ) ๐ฌ๐ข๐ง(๐ท) ๐ญ๐๐ง(๐ถ) + ๐ญ๐๐ง(๐ท) = . ๐๐จ๐ฌ(๐ถ) ๐๐จ๐ฌ(๐ท) โ ๐ฌ๐ข๐ง(๐ถ) ๐ฌ๐ข๐ง(๐ท) ๐ โ ๐ญ๐๐ง(๐ถ) ๐ญ๐๐ง(๐ท) Therefore, ๐ญ๐๐ง(๐ถ + ๐ท) = Scaffolding: Students may need to be prompted to divide the numerator and denominator by cos(๐ผ)cos(๐ฝ). ๐ญ๐๐ง(๐ถ)+๐ญ๐๐ง(๐ท) for any real numbers ๐ถ and ๐ท so that ๐๐จ๐ฌ(๐ถ) โ ๐, ๐๐จ๐ฌ(๐ท) โ ๐, and ๐โ๐ญ๐๐ง(๐ถ) ๐ญ๐๐ง(๐ท) ๐๐จ๐ฌ(๐ถ + ๐ท) โ ๐. 4. Derive a formula for ๐ฌ๐ข๐ง(๐๐) in terms of ๐ฌ๐ข๐ง(๐) and ๐๐จ๐ฌ(๐) for all real numbers ๐. Let ๐ be any real number. Then, ๐ฌ๐ข๐ง(๐๐) = ๐ฌ๐ข๐ง(๐ + ๐) = ๐ฌ๐ข๐ง(๐)๐๐จ๐ฌ(๐) + ๐๐จ๐ฌ(๐)๐ฌ๐ข๐ง(๐), which is equivalent to ๐ฌ๐ข๐ง(๐๐) = ๐ ๐ฌ๐ข๐ง(๐) ๐๐จ๐ฌ(๐). Therefore, ๐ฌ๐ข๐ง(๐๐) = ๐ ๐ฌ๐ข๐ง(๐) ๐๐จ๐ฌ(๐) for all real numbers ๐. 5. Derive a formula for ๐๐จ๐ฌ(๐๐) in terms of ๐ฌ๐ข๐ง(๐) and ๐๐จ๐ฌ(๐) for all real numbers ๐. Let ๐ be a real number. Then, ๐๐จ๐ฌ(๐๐) = ๐๐จ๐ฌ(๐ + ๐) = ๐๐จ๐ฌ(๐) ๐๐จ๐ฌ(๐) โ ๐ฌ๐ข๐ง(๐) ๐ฌ๐ข๐ง(๐), which is equivalent to ๐๐จ๐ฌ(๐๐) = ๐๐จ๐ฌ ๐ (๐) โ ๐ฌ๐ข๐ง๐ (๐). Therefore, ๐๐จ๐ฌ(๐๐) = ๐๐จ๐ฌ ๐ (๐) โ ๐ฌ๐ข๐ง๐(๐) for all real numbers ๐. Using the Pythagorean identities, you can rewrite this identity as ๐๐จ๐ฌ(๐๐) = ๐ ๐๐จ๐ฌ ๐ (๐) โ ๐ or as ๐๐จ๐ฌ(๐๐ฎ) = ๐ โ ๐ ๐ฌ๐ข๐ง๐ (๐) for all real numbers ๐. Lesson 17: Trigonometric Identity Proofs This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG II-M2-TE-1.3.0-08.2015 298 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 17 NYS COMMON CORE MATHEMATICS CURRICULUM M2 ALGEBRA II Closing (1 minute) Ask students to respond to this question in writing, to a partner, or as a class. ๏ง Edna claims that in the same way that 2(๐ผ + ๐ฝ) = 2(๐ผ) + 2(๐ฝ), it follows by the distributive property that sin(๐ผ + ๐ฝ) = sin(๐ผ) + sin(๐ฝ) for all real numbers ๐ผ and ๐ฝ. Danielle says that canโt be true. Who is correct, and why? ๏บ Danielle is correct. Given that sin(๐ผ + ๐ฝ) = sin(๐ผ)cos(๐ฝ) + cos(๐ผ)sin(๐ฝ), it follows that sin(๐ผ + ๐ฝ) = sin(๐ผ) + sin(๐ฝ) only for special values of ๐ผ and ๐ฝ. That is, when cos(๐ฝ) = 1 and cos(๐ผ) = 1, or when ๐ผ = ๐ฝ = ๐๐ for ๐ an integer. So, in general, sin(๐ผ + ๐ฝ) โ sin(๐ผ) + sin(๐ฝ). ๐ A simple example is when ๐ผ = ๐ฝ = . Then, sin(๐ผ + ๐ฝ) = sin(๐) = 0, but 2 ๐ ๐ sin(๐ผ) + sin(๐ฝ) = sin ( ) + sin ( ) = 2. Since 0 โ 2, sin(๐ผ + ๐ฝ) is generally not equal to 2 2 sin(๐ผ) + sin(๐ฝ). Exit Ticket (3 minutes) Lesson 17: Trigonometric Identity Proofs This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG II-M2-TE-1.3.0-08.2015 299 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 17 NYS COMMON CORE MATHEMATICS CURRICULUM M2 ALGEBRA II Name Date Lesson 17: Trigonometric Identity Proofs Exit Ticket Derive a formula for tan(๐ผ โ ๐ฝ) in terms of tan(๐ผ) and tan(๐ฝ), where ๐ผ โ Lesson 17: Trigonometric Identity Proofs This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG II-M2-TE-1.3.0-08.2015 ๐ ๐ + ๐๐ and ๐ฝ โ + ๐๐, for all integers ๐. 2 2 300 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 17 NYS COMMON CORE MATHEMATICS CURRICULUM M2 ALGEBRA II Exit Ticket Sample Solutions Derive a formula for ๐ญ๐๐ง(๐ถ โ ๐ท) in terms of ๐ญ๐๐ง(๐ถ) and ๐ญ๐๐ง(๐ท), where ๐ถ โ Let ๐ถ and ๐ท be real numbers so that ๐ถ โ ๐ญ๐๐ง(๐ถ โ ๐ท) = ๐ฌ๐ข๐ง(๐ถโ๐ท) . ๐๐จ๐ฌ(๐ถโ๐ท) ๐ ๐ + ๐๐ and ๐ท โ + ๐๐ , for all integers ๐. ๐ ๐ ๐ ๐ + ๐๐ and ๐ท โ + ๐๐ , for all integers ๐. Using the definition of tangent, ๐ ๐ Using the difference formulas for sine and cosine, ๐ฌ๐ข๐ง(๐ถ โ ๐ท) ๐ฌ๐ข๐ง(๐ถ) ๐๐จ๐ฌ(๐ท) โ ๐๐จ๐ฌ(๐ถ) ๐ฌ๐ข๐ง(๐ท) = . ๐๐จ๐ฌ(๐ถ โ ๐ท) ๐๐จ๐ฌ(๐ถ) ๐๐จ๐ฌ(๐ท) + ๐ฌ๐ข๐ง(๐ถ) ๐ฌ๐ข๐ง(๐ท) Dividing numerator and denominator by ๐๐จ๐ฌ(๐ถ) ๐๐จ๐ฌ(๐ท) gives ๐ฌ๐ข๐ง(๐ถ) ๐๐จ๐ฌ(๐ท) โ ๐๐จ๐ฌ(๐ถ) ๐ฌ๐ข๐ง(๐ท) ๐ญ๐๐ง(๐ถ) โ ๐ญ๐๐ง(๐ท) = . ๐๐จ๐ฌ(๐ถ) ๐๐จ๐ฌ(๐ท) + ๐ฌ๐ข๐ง(๐ถ) ๐ฌ๐ข๐ง(๐ท) ๐ + ๐ญ๐๐ง(๐ถ) ๐ญ๐๐ง(๐ท) Therefore, ๐ญ๐๐ง(๐ถ โ ๐ท) = ๐ญ๐๐ง(๐ถ)โ๐ญ๐๐ง(๐ท) ๐ ๐ , where ๐ถ โ + ๐๐ and ๐ท โ + ๐๐ , for all integers ๐. ๐+๐ญ๐๐ง(๐ถ) ๐ญ๐๐ง(๐ท) ๐ ๐ Problem Set Sample Solutions These problems continue the derivation and demonstration of simple trigonometric identities. 1. Prove the formula ๐๐จ๐ฌ(๐ถ + ๐ท) = ๐๐จ๐ฌ(๐ถ) ๐๐จ๐ฌ(๐ท) โ ๐ฌ๐ข๐ง(๐ถ) ๐ฌ๐ข๐ง(๐ท) for ๐ < ๐ถ + ๐ท < ๐ ๐ using the rectangle ๐ด๐ต๐ถ๐ท in the figure on the right and calculating ๐ท๐ด, ๐น๐ต, and ๐น๐ถ in terms of ๐ถ and ๐ท. ๐ ๐ PROOF: Let ๐ถ and ๐ท be real numbers so that ๐ < ๐ถ + ๐ท < . Then ๐ท๐ด = ๐๐จ๐ฌ(๐ถ) ๐๐จ๐ฌ(๐ท), ๐น๐ต = ๐ฌ๐ข๐ง(๐ถ) ๐ฌ๐ข๐ง(๐ท), and ๐น๐ถ = ๐๐จ๐ฌ(๐ถ + ๐ท). Because ๐น๐ถ = ๐ท๐ด โ ๐น๐ต, it follows that ๐ ๐ ๐๐จ๐ฌ(๐ถ + ๐ท) = ๐๐จ๐ฌ(๐ถ) ๐๐จ๐ฌ(๐ท) โ ๐ฌ๐ข๐ง(๐ถ) ๐ฌ๐ข๐ง(๐ท) for ๐ < ๐ถ + ๐ท < . 2. Derive a formula for ๐ญ๐๐ง(๐๐) for ๐ โ ๐ ๐๐ ๐ + and ๐ โ + ๐๐ , for all integers ๐. ๐ ๐ ๐ ๐ ๐๐ ๐ + , and ๐ โ + ๐๐ , for all integers ๐. In the formula ๐ ๐ ๐ ๐ญ๐๐ง(๐ถ)+๐ญ๐๐ง(๐ท) ๐ญ๐๐ง(๐)+๐ญ๐๐ง(๐) ๐ญ๐๐ง(๐ถ + ๐ท) = , replace ๐ถ and ๐ท both by ๐. The resulting equation is ๐ญ๐๐ง(๐๐) = , ( ) ๐โ๐ญ๐๐ง ๐ถ ๐ญ๐๐ง(๐ท) ๐โ๐ญ๐๐ง(๐) ๐ญ๐๐ง(๐) PROOF: Let ๐ be any real number so that ๐ โ which is equivalent to ๐ญ๐๐ง(๐๐) = 3. ๐ ๐ญ๐๐ง(๐) ๐ ๐๐ ๐ for ๐ โ + and ๐ โ + ๐๐ , for all integers ๐. ๐ ๐ ๐ ๐โ๐ญ๐๐ง๐(๐) Prove that ๐๐จ๐ฌ(๐๐) = ๐๐๐จ๐ฌ ๐ (๐) โ ๐ for any real number ๐. PROOF: Let ๐ be any real number. From Exercise 3 in class, we know that ๐๐จ๐ฌ(๐๐) = ๐๐จ๐ฌ ๐ (๐) โ ๐ฌ๐ข๐ง๐ (๐) for any real number ๐. Using the Pythagorean identity, we know that ๐ฌ๐ข๐ง๐ (๐) = ๐ โ ๐๐จ๐ฌ ๐ (๐). By substitution, ๐๐จ๐ฌ(๐๐) = ๐๐จ๐ฌ ๐ (๐) โ ๐ + ๐๐จ๐ฌ ๐ (๐). Thus, ๐๐จ๐ฌ(๐๐) = ๐๐๐จ๐ฌ ๐ (๐) โ ๐ for any real number ๐. Lesson 17: Trigonometric Identity Proofs This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG II-M2-TE-1.3.0-08.2015 301 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 17 NYS COMMON CORE MATHEMATICS CURRICULUM M2 ALGEBRA II 4. Prove that ๐ ๐ โ ๐๐จ๐ฌ(๐) = ๐ฌ๐ข๐ง(๐) โ ๐ญ๐๐ง(๐) for ๐ โ + ๐๐ , for all integers ๐. ๐๐จ๐ฌ(๐) ๐ We begin with the left side, get a common denominator, and then use the Pythagorean identity. PROOF: Let ๐ be a real number so that ๐ โ ๐ + ๐๐ , for all integers ๐. Then, ๐ ๐ ๐ โ ๐๐จ๐ฌ ๐ (๐) โ ๐๐จ๐ฌ(๐) = ๐๐จ๐ฌ(๐) ๐๐จ๐ฌ(๐) ๐ฌ๐ข๐ง๐ (๐) = ๐๐จ๐ฌ(๐) ๐ฌ๐ข๐ง(๐) = โ ๐ฌ๐ข๐ง(๐) ๐๐จ๐ฌ(๐) = ๐ฌ๐ข๐ง(๐) โ ๐ญ๐๐ง(๐). ๐ ๐ Therefore, โ ๐๐จ๐ฌ(๐) = ๐ฌ๐ข๐ง(๐) โ ๐ญ๐๐ง(๐), where ๐ โ + ๐๐ , for all integers ๐. ๐๐จ๐ฌ(๐) ๐ 5. ๐ ๐ ๐ ๐ Write as a single term: ๐๐จ๐ฌ ( + ๐ฝ) + ๐๐จ๐ฌ ( โ ๐ฝ). We use the formulas for the cosine of sums and differences: ๐ ๐ ๐ ๐ ๐ ๐ ๐๐จ๐ฌ ( + ๐ฝ) + ๐๐จ๐ฌ ( โ ๐ฝ) = ๐๐จ๐ฌ ( ) ๐๐จ๐ฌ(๐ฝ) โ ๐ฌ๐ข๐ง ( ) ๐ฌ๐ข๐ง(๐ฝ) + ๐๐จ๐ฌ ( ) ๐๐จ๐ฌ(๐ฝ) + ๐ฌ๐ข๐ง ( ) ๐ฌ๐ข๐ง(๐ฝ) ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ = ๐๐จ๐ฌ(๐ฝ) โ ๐ฌ๐ข๐ง(๐ฝ) + ๐๐จ๐ฌ(๐ฝ) + ๐ฌ๐ข๐ง(๐ฝ) โ๐ โ๐ โ๐ โ๐ = โ๐ ๐๐จ๐ฌ(๐ฝ). ๐ ๐ ๐ ๐ Therefore, ๐๐จ๐ฌ ( + ๐ฝ) + ๐๐จ๐ฌ ( โ ๐ฝ) = โ๐ ๐๐จ๐ฌ(๐ฝ). 6. Write as a single term: ๐ฌ๐ข๐ง(๐๐°) ๐๐จ๐ฌ(๐๐°) โ ๐๐จ๐ฌ(๐๐°) ๐ฌ๐ข๐ง(๐๐°). Begin with the formula ๐ฌ๐ข๐ง(๐ถ โ ๐ท) = ๐ฌ๐ข๐ง(๐ถ) ๐๐จ๐ฌ(๐ท) โ ๐๐จ๐ฌ(๐ถ) ๐ฌ๐ข๐ง(๐ท), and let ๐ถ = ๐๐° and ๐ท = ๐๐°. ๐ฌ๐ข๐ง(๐๐°) ๐๐จ๐ฌ(๐๐°) โ ๐๐จ๐ฌ(๐๐°) ๐ฌ๐ข๐ง(๐๐°) = ๐ฌ๐ข๐ง(๐๐° โ ๐๐°) = ๐ฌ๐ข๐ง(๐๐°). 7. Write as a single term: ๐๐จ๐ฌ(๐๐) ๐๐จ๐ฌ(๐) + ๐ฌ๐ข๐ง(๐๐) ๐ฌ๐ข๐ง(๐). Begin with the formula ๐๐จ๐ฌ(๐ถ โ ๐ท) = ๐๐จ๐ฌ(๐ถ) ๐๐จ๐ฌ(๐ท) + ๐ฌ๐ข๐ง(๐ถ) ๐ฌ๐ข๐ง(๐ท), and let ๐ถ = ๐๐ and ๐ท = ๐. ๐๐จ๐ฌ(๐๐) ๐๐จ๐ฌ(๐) + ๐ฌ๐ข๐ง(๐๐) ๐ฌ๐ข๐ง(๐) = ๐๐จ๐ฌ(๐๐ โ ๐) = ๐๐จ๐ฌ(๐) 8. Write as a single term: ๐ฌ๐ข๐ง(๐ถ+๐ท)+๐ฌ๐ข๐ง(๐ถโ๐ท) ๐๐จ๐ฌ(๐ถ) ๐๐จ๐ฌ(๐ท) , where ๐๐จ๐ฌ(๐ถ) โ ๐ and ๐๐จ๐ฌ(๐ท) โ ๐. Begin with the formulas for the sine of the sum and difference: ๐ฌ๐ข๐ง(๐ถ + ๐ท) + ๐ฌ๐ข๐ง(๐ถ โ ๐ท) ๐ฌ๐ข๐ง(๐ถ)๐๐จ๐ฌ(๐ท) + ๐๐จ๐ฌ(๐ถ)๐ฌ๐ข๐ง(๐ท) + ๐ฌ๐ข๐ง(๐ถ) ๐๐จ๐ฌ(๐ท) โ ๐๐จ๐ฌ(๐ถ) ๐ฌ๐ข๐ง(๐ท) = ๐๐จ๐ฌ(๐ถ) ๐๐จ๐ฌ(๐ท) ๐๐จ๐ฌ(๐ถ) ๐๐จ๐ฌ(๐ท) ๐ ๐ฌ๐ข๐ง(๐ถ) ๐๐จ๐ฌ(๐ท) = ๐๐จ๐ฌ(๐ถ) ๐๐จ๐ฌ(๐ท) ๐ ๐ฌ๐ข๐ง(๐ถ) = ๐๐จ๐ฌ(๐ถ) = ๐ ๐ญ๐๐ง(๐ถ). Lesson 17: Trigonometric Identity Proofs This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG II-M2-TE-1.3.0-08.2015 302 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 17 NYS COMMON CORE MATHEMATICS CURRICULUM M2 ALGEBRA II 9. Prove that ๐๐จ๐ฌ ( ๐๐ + ๐ฝ) = ๐ฌ๐ข๐ง(๐ฝ) for all values of ๐ฝ. ๐ PROOF: Let ๐ฝ be any real number. Then, from the formula for the cosine of a sum, ๐๐ ๐๐ ๐๐ ๐๐จ๐ฌ ( + ๐ฝ) = ๐๐จ๐ฌ ( ) ๐๐จ๐ฌ(๐ฝ) โ ๐ฌ๐ข๐ง ( ) โ ๐ฌ๐ข๐ง(๐ฝ) ๐ ๐ ๐ = ๐ โ ๐๐จ๐ฌ(๐ฝ) โ (โ๐) ๐ฌ๐ข๐ง(๐ฝ) = ๐ฌ๐ข๐ง(๐ฝ). Therefore, ๐๐จ๐ฌ ( ๐๐ + ๐ฝ) = ๐ฌ๐ข๐ง(๐ฝ) for all values of ๐ฝ. ๐ 10. Prove that ๐๐จ๐ฌ(๐ โ ๐ฝ) = โ ๐๐จ๐ฌ(๐ฝ) for all values of ๐ฝ. PROOF: Let ๐ฝ be any real number. Then, from the formula for the cosine of a difference, ๐๐จ๐ฌ(๐ โ ๐ฝ) = ๐๐จ๐ฌ(๐ ) ๐๐จ๐ฌ(๐ฝ) + ๐ฌ๐ข๐ง(๐ ) ๐ฌ๐ข๐ง(๐ฝ) = โ ๐๐จ๐ฌ(๐ฝ). Therefore, ๐๐จ๐ฌ(๐ โ ๐ฝ) = โ ๐๐จ๐ฌ(๐ฝ) for all real numbers ๐ฝ. Lesson 17: Trigonometric Identity Proofs This work is derived from Eureka Math โข and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG II-M2-TE-1.3.0-08.2015 303 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
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