9.1 Part 1 Notes Video

Summary

In this video, students are introduced to strategies for proving trigonometric identities from Chapter 9. The learning target aims to apply five key strategies: factoring, common denominators, expansion, substitution, and conjugates. The video revisits previous lessons, particularly Lesson 6.5, and emphasizes the use of quotient, reciprocal, negative angle, and Pythagorean identities from the formula sheet. Sara Baumgard walks through strategies including expressing trigonometric functions in terms of sine and cosine, making common denominators, reducing fractions, and identifying Pythagorean identities. Several examples illustrate these methods, demonstrating proof techniques and their practical applications. The lesson concludes with a reminder that proofs involve manipulating only one side of an equation at a time.

Highlights

• The video covers five main strategies for proving trigonometric identities: factoring, common denominators, expansion, substitution, and conjugates 💡.
• Emphasizes the transformation of trigonometric functions into terms of sine and cosine as a foundational step 🔧.
• Illustrates how Pythagorean identities simplify complex expressions, making the proof process smoother 🎯.
• Sara Baumgard provides step-by-step examples illustrating the application of these strategies in real proof scenarios 👩‍🏫.
• The lesson revisits concepts covered in Lesson 6.5, reinforcing the material with practical examples ✨.

Key Takeaways

• The importance of manipulating trigonometric identities primarily in terms of sine and cosine 🙌.
• Understanding and applying various strategies like factoring, using common denominators, and expansion to simplify trigonometric proofs 🎓.
• Emphasizing the relevance of Pythagorean identities and how they can simplify the conversion of expressions 🤓.
• Highlighting the methodical nature of proofs, reminding that only one side of the equation should be changed 🔄.
• Trigonometric proofs benefit greatly from recognizing patterns and applying identities learned in previous lessons 📚.

Overview

The video, led by Sara Baumgard, delves into the strategies necessary for proving trigonometric identities, a key component of Chapter 9. Students are encouraged to revisit prior lessons, especially Lesson 6.5, to fortify their understanding of quotient, reciprocal, and negative angle identities. The lesson is interactive, engaging students with thoughtful examples and breaking down complex problems into manageable steps.

A variety of strategies are presented as tools for simplifying and proving trigonometric identities, enhancing students' problem-solving toolkit. Sara highlights the importance of expressing functions primarily in terms of sine and cosine, reducing fractions, and using identities like the Pythagorean to transform expressions. This approach is designed to demystify the proof process, making it more accessible and less daunting for students.

Throughout the lesson, several examples are dissected to demonstrate the application of these strategies in real-world proofs. From changing expressions with common denominators to employing expansion techniques, students get a comprehensive look at how these methods interplay. The video closes with a reminder that proofs require changing only one side of an equation, cementing the systematic methodology needed for success in this subject area.

Chapters

• 00:00 - 00:30: Introduction to Chapter 9 Chapter 9 focuses on applying strategies to prove trigonometric identities. There are five main strategies discussed: factoring, common denominators, expansion, substitution, and conjugates. Some identities were also covered in lesson 6.5. Key identities mentioned include the Pythagorean identities, reciprocal identities, and negative identities.
• 00:30 - 01:00: Basic Trigonometric Identities The chapter titled 'Basic Trigonometric Identities' covers fundamental trigonometric identities, focusing initially on quotient identities. The lesson revisits content from a previous lesson (6.5), emphasizing the identities: tangent (tan) being sine over cosine and cotangent (cot) being cosine over sine. These quotient identities are highlighted as useful in the context of the chapter. The chapter also acknowledges reciprocal identities, though details of those are not included in the provided transcript.
• 02:30 - 04:00: Writing Sine in Different Forms This chapter discusses converting sine into different forms using trigonometric identities. It begins by revisiting the definitions of six fundamental trigonometric identities and emphasizes the utility of reciprocal identities. The chapter also highlights the importance of recognizing and using negative angle identities and periodical identities, which involve adding or subtracting 2π to find equivalent sine values.
• 04:30 - 06:30: Strategies for Trigonometric Proofs The chapter focuses on strategies for proving trigonometric identities. It highlights similar identities for different trigonometric functions like cosine, cosecant, and tangent. A key point noted is that for tangent and cotangent, c is added or subtracted in certain proofs. The chapter also covers negative angle identities, such as how sine of a negative angle becomes the negative of sine, whereas for cosine the negative sign disappears.
• 07:00 - 10:00: Example 1: Factoring The chapter titled 'Example 1: Factoring' revisits basic trigonometric identities, including the sine and tangent properties when dealing with negative angles. It emphasizes the importance of understanding Pythagorean identities and their variations, suggesting that it might be useful to refer to chapter six notes, especially lesson 6.5, which covers these topics in detail. The discussion points to the flexibility in writing trigonometric functions.
• 10:00 - 18:00: Example 2: Common Denominators The chapter explores different ways of expressing sine of x. The first method is not visible in the transcript, but it suggests the definition of sine as 1 over cosecant, using the reciprocal identity. The second method involves a discussion that requires further elaboration not provided in this partial transcript.
• 18:00 - 27:00: Example 3: Expansion This chapter discusses the characteristics of a sine graph, mentioning key points on the graph such as pi over 2, pi, 3 pi over 2, and 2 pi. It notes that the sine graph extends up to 1 and down to -1. Furthermore, the chapter explains that a sine graph can be expressed as a cosine graph with a phase shift to the right by pi over two, highlighting the relationship between sine and cosine functions.
• 27:00 - 30:00: Example 4: Substitution The chapter discusses the concept of substitution in trigonometry, focusing on transforming sine graphs into cosine graphs through phase shifts. An example given is the cosine graph derived from a sine graph with a phase shift of pi over 2. The chapter also covers the negative angle identity, explaining how the negative sign can be factored out of the angle measurement.
• 30:00 - 30:30: Conclusion The chapter titled 'Conclusion' emphasizes the importance of understanding the different ways to write the sine function. It suggests thinking of these three variations and introduces nine strategies to assist with trigonometry proofs. It indicates that these strategies are not ranked in order of importance but serve as general guidelines to help solve trig proofs.

9.1 Part 1 Notes Video Transcription

• 00:00 - 00:30 hi students today we are starting chapter nine uh our learning target states that i can apply strategies to prove trigonometric identities there are five strategies that we will discuss and those being factoring common denominators expansion substitution and conjugates um some of these identities that we're going to be talking about today we also talked about in lesson 6.5 and there are several identities on your formula sheet so what i mean by that is like the pythagorean your reciprocal identities your negative
• 00:30 - 01:00 angle identities um those sorts of things so just kind of keep an eye on that in your formula sheet all right so our basic trig identities what i kind of talked about is so we first when we learned these in lesson 6.5 it was quotient identities so we first talked about tangent and the definition of tangent being sine over cosine and then the definition of cotangent being cosine over sine so two quotient identities there that will be useful here and then we have reciprocal
• 01:00 - 01:30 identities so these six um just the definitions of them and then we learned that in lesson 6.5 sometimes it is useful to convert a reciprocal identity and to use your reciprocal identities all right one other thing to be aware of is your negative angle identities and your periodical identity so periodical identity that is where we took sine and you would always add or subtract 2 pi in order to find
• 01:30 - 02:00 an identity that is similar to the one that's given to us same thing with cosine and cosecant and tangent but then the difference here remember with tangent it in cotangent it is pi that you are adding or you are subtracting okay negative angle identities sine of negative x can also be written as negative sine of x cosine of negative x the negative goes away
• 02:00 - 02:30 and tangent of negative x the negative gets brought out front okay and then more basic trigonometric identities are pythagorean identities and then remember that there are variations of each right and we had that chart so um it could be a good idea to have your chapter six notepackets out in front of you in lesson 6.5 where we talked in detail about this so trig functions can be written in many
• 02:30 - 03:00 different ways we are going to write sine of x in three different ways now yes this is here but i want to talk about them so write sine of x in three different ways so this is what you see let's pretend we don't see this first one all right well what's the definition of sine the definition of sine we can also write it as 1 over cosecant that reciprocal identity the second one here the definition of sine now this one requires us to talk about a
• 03:00 - 03:30 sine graph okay so we have pi over 2 pi 3 pi over 2 2 pi we're going up to 1 and down to negative 1. and we have a graph that looks something roughly like this okay a sine graph can also be written as a cosine graph with the shape phase shift to the right pi over two because look at right here at pi
• 03:30 - 04:00 over 2 we have a cosine the beginning of a cosine graph so from that normal sine graph we could get a cosine graph with a phase shift of pi over 2. and then the last one here this third one the negative angle identity remember with this we took the negative and you brought that negative out of the sign or out of that x and then that negative
• 04:00 - 04:30 times the negative is what gives you the positive sign okay so three different ways to write sine and it's always good to kind of be thinking of these three different ways so strategies for trig proofs they're not necessarily in this order of like what's most important or not but these are just nine strategies that can help you with trig proofs number one
• 04:30 - 05:00 represent everything in terms of or express everything so think about this when we took tangent what did we do to it we turned that into sine over cosine so often it's a really good idea to express everything in terms of sine or cosine second thing make a common denominator
• 05:00 - 05:30 that was always a good idea for us third thing a fraction within a fraction so for example if we had tangent and we simplified that to sine over cosine let's say we had tangent divided by cotangent well cotangent is cosine over sine so that is a fraction within a fraction and then we can multiply by the reciprocals fourth idea is to reduce fractions
• 05:30 - 06:00 always a good idea to reduce we know that fifth one what or what foil or factor when you look at those trig proofs can you factor it or should you foil it should you distribute six look for pythagorean identities
• 06:00 - 06:30 so for example when we saw something that was like 2 cosine this is just an example 2 cosine squared x plus sine x minus 3 something like that well we can't solve that right now right so we would take this cosine squared and we would think pythagorean identity we could change that to 1 minus sine squared x and then that could potentially be
• 06:30 - 07:00 something we could factor but then we'd foil this and so forth okay seventh thing use conjugates use conjugate so um for example like using the opposite we talked a lot about conjugates and when we were dealing with imaginary numbers eighth one separate a fraction and then the last one multiply by one okay so just some examples there some strategies uh we have four
• 07:00 - 07:30 examples here that we are going to talk about first one factoring so this is an example here prove that sine cubed plus sine x times cosine squared equals sine of x remember with proofs you only change one side do not change the other so clearly we need to change this left side well a couple different things we could do here
• 07:30 - 08:00 you could see this cosine squared and you could think hmm i got to change that to a pythagorean identity so that is a method that could work also we could think about foiling but like kind of the opposites like you're dividing out what you uh what these share and a sine cubed plus sine x times cosine squared what do they share they share a sign so let's take a sign out so if i divide a sign out sine cubed divided by sine is sine
• 08:00 - 08:30 squared and then i'm dividing out that sine so the sine is out front so then what i'm left with is just a cosine squared okay well now pythagorean identities like stop right here what is that what is sine squared plus cosine squared so hopefully what we are thinking is that is one so then look at that you have 1 times your sine of x equals sine of x yes that's true so last
• 08:30 - 09:00 step just say sine of x equals sine of x done okay this one's a little bit longer but now we have a second example and this is called common denominators so you can see here there's some things that we're going to have to change here um to get all of this to equal 2 cosecant x well what common denominators means we have to multiply either side of the fraction by the denominator of
• 09:00 - 09:30 the other fraction so that means that we need to take 1 plus cosine x and we need to multiply by the denominator sine of x but what you do to the bottom you have to do the top this one we need to multiply by one plus cosine x on the top and the bottom okay well we have 1 plus cosine x times another 1 plus cosine x
• 09:30 - 10:00 so we'll end up foiling that and then we have sine x times the sine x well that's sine squared we can combine that all over one plus cosine x times sine of x okay well let's take this 1 plus cosine x this guy and let's foil it so if we foil that we have 1
• 10:00 - 10:30 plus we have cosine x plus a cosine x so that's 2 cosine x plus cosine squared x plus sine squared x all over 1 plus cosine x times sine of x okay well we're kind of done here there's not much else we can do but look at this cosine squared plus sine squared
• 10:30 - 11:00 and what is that that is just one because of the pythagorean identity so don't just cross it out that means one so then you have one plus two cosine x plus one so this becomes 2 plus 2 cosine x all over 1 plus cosine x times sine of x okay again we're kind of stuck here
• 11:00 - 11:30 you take a look at the top and you ask yourself is there something that they share okay well yes they do they share a two so let's see if we can take that two out so if i take the two out of the top then i'm left with 1 plus 1 cosine x and look at that something can cross out on the bottom remember you can only cross out when you are multiplying so this goes away
• 11:30 - 12:00 this goes away and what we're left with is 2 divided by sine of x which is 2 cosecant x and that's what we are trying to prove so last step show that the left side in fact equals the right all right fourth one or third one here expansion
• 12:00 - 12:30 okay these are going to get a little bit trickier but you got it so expansion here expansion think about that meaning foiling so we're going to have to foil this out that means this gets multiplied by 1 cosecant multiplied by cosine cotangent by 1 cotangent by cosine okay so we have cosecant x
• 12:30 - 13:00 minus cosecant x times cosine x plus cotangent x minus cotangent x times cosine x okay so you're thinking like why do you do that um what does that do we'll go back to the very first strategy for trig proofs and their very first strategy when we get to this point is let's make everything sine and cosine and then maybe some things can start combining
• 13:00 - 13:30 all right we're making something sine and cosine so what is the definition of cosecant cosecant means one over sine all right cosecant is the definition of one over sine again times cosine so that just stays up top plus cotangent cosine over sine minus cotangent so cosine over sine times cosine on top
• 13:30 - 14:00 okay well this is really nice because it looks like there's common denominators here so let's combine these um into fractions so think about this as kind of one let's combine that into one fraction can we combine this into one fraction and then make it all one so that would be one minus sine x minus cosine x over sine x you can see what's going to happen there
• 14:00 - 14:30 plus again cosine x all over sine x minus cosine times cosine so cosine squared x all over sine x all right we have 1 minus cosine x all over sine x so that's combining these two fractions right here let's combine these two right here so that becomes minus or plus
• 14:30 - 15:00 cosine x minus cosine squared x all over sine x okay we can now make this again one big fraction um i'm gonna do that right down here so that if we can make that one big fraction that's 1 minus cosine x plus cosine x
• 15:00 - 15:30 minus cosine squared x all over sine x now look at these cosines negative cosine plus cosine those go away so what we're left with is 1 minus cosine squared x all over sine x okay now you're kind of done but pythagorean identity what is another way to think about one
• 15:30 - 16:00 minus cosine squared that is another way to write sine squared x and we still have that all of our sign so one of our signs cancel and we're left with sine um oh last thing here show left side equals the right okay last one here
• 16:00 - 16:30 prove that this fraction equals the left now you could kind of change either but the left here has a lot more going on so let's let's change that so that it equals the right then well rewriting this as 1 minus sine squared so 1 minus sine squared so i'm just kind of grouping these two things together
• 16:30 - 17:00 plus sine of x so i haven't changed anything all over cosine x all right well this method is called substitution so substitution we just did it what is another way to write one minus sine squared well now one minus sine squared we can write as cosine squared
• 17:00 - 17:30 now you might be thinking like how does that help me look look down at method number eight method number eight said separate your fractions so we could separate these fractions to saying cosine squared all over cosine plus sine over cosine okay well cosine squared over cosine is just cosine
• 17:30 - 18:00 and sine over cosine is tangent and look at that that's what we were trying to prove on the right side so there you go that equals cosine x plus tan of x [Applause] all right you guys um and just remember that a proof is not an equation so an equation meaning like subtract to one side or add to the other you're only only changing one side
• 18:00 - 18:30 okay that is uh lesson 9.1 part 1. have a good day