PrecalcB 2.4

Summary

In this lesson, Danielle Nazaroff dives into unit 2 lesson 4 of PrecalcB, focusing on transforming trigonometric functions using sum-to-product and product-to-sum identities. The session involves rewriting products of trigonometric functions into sums or differences and vice versa. With a step-by-step approach, Danielle demonstrates through several examples, how to simplify expressions such as sine and cosine products or sums by applying these identities. The lesson emphasizes the importance of being familiar with these transformations for solving trigonometric problems efficiently.

Highlights

• Introduction to sum-to-product and product-to-sum identities for trigonometric functions. 🎓
• Step-by-step examples to simplify trig expressions using these identities. ✏️
• Tips on utilizing even and odd identities to manage negative angles. 🔄
• Insights on factoring and distributing in complex trig equations. 🔍
• Practicalities of converting between sums and products to find exact values. 📐

Key Takeaways

• Transform products into sums or differences for easier trigonometric simplification! 🌟
• Master sum-to-product identities to handle complex trig expressions like a pro! 📘
• Practice makes perfect—use cheat sheets for quick reference during exams! 📝
• Even and odd identities are your friends; use them wisely! 🤓
• Identities might look similar, but each has a unique twist—stay sharp! ⚡

Overview

Danielle opens with a warm welcome and introduces the topic of sum-to-product and product-to-sum identities. These are essential tools for converting between products and sums in trigonometric expressions. Her detailed walkthrough provides a roadmap for efficiently dealing with trigonometric identities throughout the lesson.

Throughout her insightful teaching, Danielle emphasizes the importance of understanding these identities. She helps students overcome the visual similarity of different trigonometric identities by employing a systematic approach to simplify expressions. Examples are tackled methodically, ensuring comprehension.

The lesson concludes with practice problems, proving identities, and finding exact values, solidifying the utility of these identities. Danielle encourages students to create cheat sheets and utilize a structured approach to recognizing patterns in identities, ensuring they are well-equipped for quizzes and tests.

Chapters

• 00:00 - 00:30: Introduction to Lesson 2.4 The chapter titled 'Introduction to Lesson 2.4' begins with a welcome message, indicating a continuation from previous lessons. It focuses on Unit 2, Lesson 4, which covers the mathematical concepts of sum to product and product to sum identities. The emphasis is on adding more identities to students' knowledge base and suggests that students may want to note these down on a cheat sheet for easier reference.
• 00:30 - 01:00: Sum to Product and Product to Sum Identities The chapter introduces the sum-to-product and product-to-sum identities in trigonometry.
• 01:00 - 02:00: Example 1: Multiplying Trig Functions In this chapter titled 'Example 1: Multiplying Trig Functions', the focus is on multiplying trigonometric functions using identities. The example provided involves S(3x) * cos(2x) and demonstrates the substitution and simplification steps by using addition and subtraction identities, specifically breaking it down to S(3x + 2x) + S(3x - 2x). The process emphasizes organizing and 'cleaning up' the components into a simplified form as part of solving the expression.
• 02:00 - 03:00: Example 2: Use of Even and Odd Identities In this chapter titled 'Example 2: Use of Even and Odd Identities', the focus is on working through mathematical problems involving sine functions. The discussion includes manipulating trigonometric expressions such as sin(5x) + sin(x) and distributing constants within these expressions. Additionally, the problem involves analyzing products like sin(4θ) * sin(7θ), emphasizing techniques in using even and odd identities to simplify or resolve the equations. Through practical examples, this chapter aims to enhance understanding of how these trigonometric identities can be applied to solve problems effectively.
• 03:00 - 04:00: Example 3: Factoring and Simplifying This chapter focuses on the concepts of factoring and simplifying in the context of mathematical expressions. It covers techniques for rewriting and simplifying expressions, possibly using trigonometric identities. In the provided example, the process involves working with angles and trigonometric functions, demonstrating how to clean up and simplify the expression through factoring and combining like terms.
• 04:00 - 05:00: Example 4: Exact Values with Angles This chapter discusses the calculation of exact values for trigonometric functions involving angles, specifically focusing on the use of even and odd identities. It highlights the property of the cosine function being even, which allows the equivalence of cosine of a negative angle to the cosine of its positive counterpart. The explanation uses 'three theta' as a case study to illustrate this property.
• 05:00 - 07:00: Example 5: Dealing with Denominators Chapter Title: Example 5: Dealing with Denominators Summary: The chapter discusses mathematical operations involving trigonometric functions and denominators. It explains the simplification of expressions, specifically focusing on using cosine functions. An example is provided demonstrating the rewriting of an expression with a cosine function and distributing a constant across it. The process is illustrated through specific numeric values like 12, 3 Alpha, and 11 Theta. Another problem involves solving with -14 cosine Alpha * cosine 5 Alpha. The approach involves systematic simplification of trigonometric expressions.
• 07:00 - 09:00: Converting Sum or Difference to Product The chapter focuses on the technique of converting sums or differences into products. It begins with factoring out a constant, in this case, 14. The approach then involves dealing with the product of cosine terms, specifically analyzing a particular example. This chapter likely delves further into the mathematical procedures and concepts necessary for such conversions.
• 09:00 - 11:00: Example 6: Simplifying with Negative Angles In this example, the speaker begins with an expression that includes both cosine and Alpha terms and aims to simplify it, particularly focusing on handling negative angles and their implications within the mathematical context. The initial expression is broken down and recalculated through steps, involving multiplication and simplification processes to achieve a more manageable form, exploring the effects of negative and positive angles on the final results.
• 11:00 - 14:00: Example 7: Evaluating Trigonometric Functions The chapter discusses the evaluation of trigonometric functions, specifically focusing on sine and cosine. It involves transformations such as using even and odd identities to handle negative angles and multiple angles, like -4 alpha and 6 alpha. The process includes distributing terms during evaluation to simplify expressions.
• 14:00 - 20:00: Proving Trigonometric Identities The chapter discusses methods and strategies for proving trigonometric identities. It begins with a series of examples that illustrate how to find exact values of trigonometric functions, in this case focusing on the cosine function. The chapter aims to build on foundational trigonometric knowledge and enhance problem-solving skills through practical demonstration.
• 20:00 - 20:30: Conclusion and Encouragement In the conclusion chapter, the focus is on reinforcing the application of trigonometric identities, specifically the cosine and sine functions. The dialogue becomes technical, likely addressing the calculations or formulas previously discussed. Although fragmented, the attempt is to solidify learners' understanding and encourage precise application of the mathematical concepts. This section serves to summarize and encourage further inquiry into these techniques, despite the challenges met along the learning journey.

PrecalcB 2.4 Transcription

• 00:00 - 00:30 hello everyone and welcome back we are looking at unit 2 lesson four which is going to look at our sum to product and our product to Su identities so you can already see them right there at the top um we're just adding on more and more identities for this unit so definitely something that you might want to like jot down on a little cheat sheet for
• 00:30 - 01:00 yourself or a note card just because there's a lot of them they all start to look a little similar um but yes these ones are going to basically take us with um starting off here with just if we're multiplying two trig functions how we can write them to be a sum or difference and then of course in our next set of examples we'll go the other direction taking sums and differences and writing them as products all right so diving into these examples um
• 01:00 - 01:30 looking at the first one I have S of 3x * cosine of 2x so for that one um I'll be using this identity here at the top so I'm going to write it as 12 of s of 3x + 2x+ S of 3x - 2x we'll clean everything up so 12
• 01:30 - 02:00 sin of 5x + S of X and then distribute that 12 all right so that's going to be that one and then looking at number two sin of 4 thet * sin of 7 Theta so if I was to look at that one I'm going to be looking at this
• 02:00 - 02:30 identity here so I'm going to rewrite this as 12 * cine of 4 th - 7 thet - cosine of 4 Theta + 7 Theta just cleaning everything up
• 02:30 - 03:00 right you can see um on the left with that cosine of -3 alpha or excuse me3 Theta we can use our even and odd identities so we know that cosine is even which means if I do a little thinking Cloud off to the side here cosine of a negative angle so in this case-3 Theta would be equivalent to just the regular all positive cosine three Al 3 Theta that's what it means for it to be an
• 03:00 - 03:30 even function um so we can rewrite that to just be 12 * cosine 3 Alpha thetus cosine of 11 thet and then distribute that 1 12 all right number three we have -14 cosine Alpha * cosine 5 Alpha um what
• 03:30 - 04:00 I'm going to do with this I'm going to first basically factor out that 14 and now I'll deal with my product so I'm multiplying to the cosin so I'm going to look at um this one right here
• 04:00 - 04:30 all right so I got4 * 12 cosine Alpha - 5 Alpha plus cosine Alpha + 5 Alpha there we go I think I got it all closed up now um I'm going to multiply
• 04:30 - 05:00 the -14 and the 1/2 so that'll be -7 and then cine of -4 Alpha plus cosine of 6 Alpha and then again with another negative inside with using my even and odd this is even so cine 4 Alpha plus cine 6 Alpha and distributing at
• 05:00 - 05:30 -7 all right so now um let's look at our next set of examples so we want to find the exact values um for um for these so I have cosine of
• 05:30 - 06:00 195° * s of 135° so um if we go back up here it's um a cosine times a sign so cosine times the sign means I'm using this one right here so I'm going to write this out 12 s of
• 06:00 - 06:30 195° + 135° - s of 195° minus 135° okay so 195 + 135 is 330 and then 195 - 13 [Music]
• 06:30 - 07:00 60 so s of 60° all right so s of 330 is going to be negative a half and then s of 60 is going to be R 3 over 2 you could distribute the 1/2 now so /4 - R 3 4
• 07:00 - 07:30 and write this as that all right looking at number five with the8 cosine of piun 4 * cosine of P 12 um again with the problem similar to this just uh previously we'll factor out the8 first and then deal with my product so I'll have2 um with the double cosine this
• 07:30 - 08:00 will just be um cine < 4 - > / 12 plus cine pi over 4 + piun / 12 all right multiply the8 and the one2 I will need to get common denominator so that'll become 3 pi [Music]
• 08:00 - 08:30 12 that's 2 piun 12 Pi 6 here this will become 3 piun over 12 + < / 12 so 4 piun 12 or pi over 3 so now you just know kind of ahead of time so cosine ofk 6 plus cosine ofun 3
• 08:30 - 09:00 all right cosine of pi/ 6 um is going to be R 3/ 2 plus cosine of pi/ 3 is 12 then distribute here so -2 R 3 - 2 all right so moving on now to taking a
• 09:00 - 09:30 product or excuse me a sum or difference and writing it as a product so you can see our identities there at the top so it says here in the directions to write as a sum or difference but that's actually supposed to say write as a product excuse me all right so um we have cosine of 8X plus cosine of
• 09:30 - 10:00 2x um so I am going to write this with using this one right here so 2 cosine of 8x + 2x all 2 * cosine of 8 x - 2x all 2 simplify that's 10 x over 2 or 5
• 10:00 - 10:30 x 8 x - 2 is 6 x so 3x and we leave our answer just like that number seven s of 3 beta minus s of 6 beta so I'm just have the double signs here so that one right there so 2 cosine of 3 B beta
• 10:30 - 11:00 + 6 beta there we go cosine sign okay making sure I got my trig functions down right all right so have 2 cosine of 9 bet / 2 * s of what is that 3 beta/
• 11:00 - 11:30 2 again with the negative angles thinking about our even and odd identities s of a-3 beta/ 2 isal to S 3 beta/ 2 so let's replace that so I'll have -2 cosine of 9 beta / 2 * sin
• 11:30 - 12:00 of 3 beta/ 2 so again we just took that negative and multiplied it to the -2 in the front and I think that's as fully simplified as we can get all right last one in this set -10 cosine 3 Theta + 10 Cine 5 Theta I'm going to factor out the -10
• 12:00 - 12:30 now I can deal with my difference I'm subtracting um so I'll have um -2 s of 3 Alpha + 5 uh not Alpha theta 3 e Theta + 5 Theta / 2 * s of 3 theta plus no [Music]
• 12:30 - 13:00 minus all over 2 all right -10 * -2 is POS 20 get S of 8 Theta over 2 so 4 Theta here I get -2 thet over 2 so Theta again even and odd sign is odd so that negative is going to come out in front so -2
• 13:00 - 13:30 s of 4 Theta sin OFA all right finding the exact value um I'm going to keep that three halves out and I'll deal with this so I'm adding the two signs um so I'm going to use 2 * s of
• 13:30 - 14:00 345 + 75 / 2 time cine of 345 - 75 all over 2 345 + 75 divid half is 2 uh 10 345 - 7
• 14:00 - 14:30 half is 135 so cosine 135 If I multiply these two together they're just going to cancel out the two so it leaves me with 3 S of 210 cosine of 135 so s of 210 that's going to be my third quadrant so 12
• 14:30 - 15:00 and then cosine of 135 is in the second quadrant so that's negative R 2 over two so here I'll get um I have a double negative so positive so 3es * R 2 over 2 so 3 R 2 over 4
• 15:00 - 15:30 all right number 10 cosine of 11 piun 12us cosine of 5 piun 12 all right so I'm going to have -2 * sin 11 piun / 2 + 5 pi over 12 I think I said over two before over 12 uh divide by two that's what I'm thinking times
• 15:30 - 16:00 sign now the difference okay now let's clean it up so we can simplify so already have common denominators which is nice so we get 16 pi over 12 that can be reduced um I think reduced by four so 4 pi over 3
• 16:00 - 16:30 all right and then here I'll get when I subtract 6 piun over 12 which is really just pi/ 2 over two all right so again we have 4 pi over 3 / 2 which is the same as multiplying by the reciprocal so multiplying by 12
• 16:30 - 17:00 there's that okay and then um Pi / 2 * a half give you pi over 4 all right now we can evaluate these s of um 2 piun over 3 is going to be
• 17:00 - 17:30 23 is in my second qu so positive sign so r two R 3 over two and then s of Pi 4 is going to be R 2 over 2 all right let's just clean this up so those will cancel leaving us with < TK 6 all over
• 17:30 - 18:00 2 all right last question for this um set proving this identity again we just want to do as many examples um as we can so we can continue to look for patterns recognizing things just to help us when we are doing our own identity um questions like on our quiz or test so again we've talked about always
• 18:00 - 18:30 starting on the messy side first the stuff with a lot the side with a lot of stuff so I'm going to do the left hand side with that cosine of 4T minus cosine of 2T all over s of 4T plus s of 2T um so since I am subtracting those cosiness I'm going to write this as um2 s of 4 t + 2 T all over
• 18:30 - 19:00 2 * s of 4 t - 2T / 2 all over with those I'm going to have 2 s of 4 t + 2T all 2 * cosine of 4 t - 2T all over 2 and again this is all equal to Tang
• 19:00 - 19:30 T TW are going to cancel out which is nice so negative negative um s of 4 t + 2 T to 6t which we can reduce make that 3
• 19:30 - 20:00 T Time s of 4 t - 2 T which is 2T so just T all over s of 4 t + 2 t/ 2 so that's 6t over 2 which is 3 T time cosine of 4T - 2T is 2T / 2 is just T so these are going to end up canceling out which is nice leaving us with just the negative
• 20:00 - 20:30 get S of T over cosine of [Music] T so there we go all right as always if you have any questions concerns about anything please feel free to reach out um thank you so much for being here with us and I will see you guys all in our next lesson thank you and take care