yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Definite integral of sine and cosine product


3m read
·Nov 11, 2024

We're in our quest to give ourselves a little bit of a mathematical underpinning of definite integrals of various combinations of trig functions, so it'll be hopefully straightforward for us to actually find the coefficients, our 4A coefficients, which we're going to do a few videos from now.

We've already started going down this path. We've established that the definite integral from 0 to 2 pi of s of Mt DT is equal to zero and that the cosine, the definite integral of cosine Mt DT is equal to zero for any nonzero integer and M.

Actually, we can generalize that a little bit for sine of Mt; it could be for any M actually. And if you don't believe me, I encourage you to... So let me write this for any integer M. This top integral is going to be zero, and this second integral for any nonzero integer M...

You could see if you had zero in this second case, it would be cosine of 0 t, so this would just evaluate to one. So you'd just be integrating the value one from 0 to 2 pi, and so that's going to have a nonzero value.

So with those two out of the way, let's go a little bit deeper, get a little bit more foundations. So I'm now I now want to establish that the definite integral from 0 to 2 pi of s of Mt times cosine of NT DT, that this equals zero for any integers M and N. They could even be the same M; they don't have to necessarily be different, but they could be different.

How do we do this? Well, let's just rewrite this part right over here, leveraging some trig identities. And if it's completely unfamiliar to you, I encourage you to review your trig identities on Khan Academy.

So this is the same thing as a definite integral from 0 to 2 pi of s of Mt multiplied by cosine NT. We can rewrite it using the product-to-sum formulas. So let me use a different color here.

So this thing right over here that I've underlined in magenta, or I'm squaring off in magenta, that can be rewritten as 1/2 times s of m + n t sine of m + n t plus s of m minus n t. And then let me just close that with a DT.

Now, if we were to just rewrite this using some of our integral properties, we could rewrite it as... So this part over here... We could, and let's assume we distribute the 1/2, so we're going to distribute the 1/2 and use some of our integral properties.

And so what are we going to get? So this part roughly right over here we could rewrite as 2 times the definite integral from 0 to 2 pi of sine of m + n t DT. And then this part, once you distribute the 1/2 and you use some integral properties, this could be plus 1/2 times the definite integral from 0 to 2 pi of s of m minus n t DT.

Now, what are each of these things going to be equal to? Well, isn't this right over here? Isn't that just some integer? If I take the sum of two arbitrary integers, that's going to be some integer, so that's going to be some integer, and this two is going to be some integer right over here.

And we've already established that the definite integral of s of some integer times T DT is zero. So by this first thing that we already showed, this is going to be equal to zero. That's going to be equal to zero; it doesn't matter that you're multiplying by 1/2.

1/2 times 0 is 0, and 1/2 times 0 is 0; this whole thing is going to evaluate to zero. So there you go, we've proven that as well.

More Articles

View All
Snapchat Q&A Part 2: Commercial vs Residential Real Estate - which one is better?
I know what it’s like when you first start and you see this. It’s basically like you’re at the bottom of the mountain. You look at the very top and you’re like, “How could I get to the top of that mountain? What do I do?” It’s really overwhelming to see t…
27 Years Old: Should I buy a House or a Lamborghini?
What’s up you guys, it’s Graham here. So this is one of these things that, you know, I was pretty well set on getting a Lamborghini until I got the money to get the Lamborghini, and now I’m just like, it’s not the smartest thing to do. Are you sure about …
Why You’ll Regret Buying A Home In 2023
What’s up, guys? It’s Graham here. So given what’s happening in the housing market and the sudden decline across pretty much everything, I felt like it would be appropriate to address everyone’s concerns, share my thoughts about what’s going on, and expla…
Space Archaeology: A New Frontier of Exploration | National Geographic
(light ethereal music) We are the detectives of the past. And we have to figure out what happened. That is what is fascinating about archaeology. Peru is super special archaeologically because this is one of the cradles of civilization. It’s where civili…
Oceans 101 | National Geographic
Oceans cover over 70 percent of the Earth’s surface. They not only serve as the planet’s largest habitat, but also help to regulate the global climate. The ocean is a continuous body of salt water that surrounds the continents. It is divided into four ma…
Crossing the Creek - Deleted Scene | Life Below Zero
So I came down, I’m crossing the valley, and I want to go up on this mountain here next. This is another mountain I’m thinking about hunting on, and I want to see if there are any sheep up there today. I’m just going to have to cross this little creek her…