 In this video, I wanna demonstrate a very useful technique in calculus that's known as trigonometric substitution. As the name suggests, this is a purely trigonometric technique and it relies on a very basic trigonometric observation, basically just using the Pythagorean identity to turn a difference of squares or sum of squares into a perfect square to allow you to simplify. So these trigonometric substitutions look like the following. You're gonna take a square root of a sum of squares or a difference of squares. In this example, I'm gonna take the square root of x squared plus nine and we're gonna use the substitution that x equals three tangent theta. Now notice that x and theta are different symbols. These are different variables. So you started off with this algebraic variable x. We're gonna substitute instead this trigonometric variable theta. It's kind of like translating from a language. Like you wanna translate from French to German, let's say or you wanna translate from Korean to Japanese, what have you. Now the idea is while the same thing could be expressed in two different languages, it could be the case that the other language can more simply express that term than the language you're describing. And we see this all the time. Like if you take English and Japanese, for example, like the word sushi is a Japanese word we use in English. Why, why do we use it? And the idea is because in English, there was no word that properly described this Japanese cuisine. So we just transliterated the word. We just took the word from Japanese and now became an English word, right? And because that was the easiest way to describe it. Perhaps the other language is better in that regard. This is just an example using vocabulary. If you start throwing things like grammar and syntax into the mix, colloquial language, idioms, what have you. Language is a very complicated thing. It could be that a language that there's an easier way to explain that expression in the other language, right? I myself lived in Korea for a while and I did speak a little bit of Korean at the time. And so I was living with other people who could speak English and Korean. And we have these really interesting bilingual conversations, which are part in English, part Korean, because we would adapt our language based upon what was easier to describe. Sometimes it was easier to describe things in English. Sometimes it was easier to describe things in Korean. And in the same single sentence, we sometimes would use both Korean and English language simultaneously. And it's kind of really weird if you listen to it, but because both the speaker and the listener knew the two languages, we could use the advantages of the two languages to help us out here. I imagine this happens in other bilingual situations all the time. Maybe English and Spanish is another good example. Our two languages in play here are English and, it's not English, I'm sorry, algebra and trigonometry. We wanna use the advantages of trigonometry to simplify the algebra. So let's get into the mathematics now, allegory set aside. If you take the square root of x squared plus nine, we're gonna make the substitution. We're gonna replace the x with three tangent theta, because that's what we were told. Don't worry about why that's the substitution. That'll come much, much later. Right now we just wanna simplify the expression. Now, if you have a three tangent theta squared, this is the same thing as nine tangent squared theta plus nine. You'll notice that both terms are divisible by nine. I'm gonna factor out the nine. So we get nine times the tangent squared theta plus one. And now this is where the trigonometric identity comes into play. Recall that one plus tangent squared theta is equal to secant squared theta. This is a child of the mother Pythagorean identity. If we make this substitution, we could substitute tangent squared plus one with secant squared. And this gives us then the square root of nine times secant squared theta. For which case, oh, nine is a perfect square. The square root of nine is three. Secant squared, it's obviously a perfect square. Its square root will be secant. That's not exactly true. It's gonna be the absolute value of secant, because secant could be positive or negative depending on the quadrant. So without any information about the quadrant, we would just say three times the absolute value of secant. Although in the context in calculus, which you would use this technique, for the most part you could assume the quadrant is the first quadrant. Basically, you can basically drop the absolute value with how much problems. But without that context, we will keep it up just to be proper. Let's see another example. This time let's do a difference of squares. We have the square root of 16 minus x squared. We're gonna use the trigonometric substitution x equals four sine. All right, why are we gonna do that? Well, it turns out it's the perfect substitution to use here. 16 minus x squared inside the square root. This will become the square root of 16 minus four sine squared theta. So, excuse me, sine of the fourth, four sine, the whole thing is squared there. So then when you square that, you're gonna get 16 minus 16 sine squared. So you can start to see like, okay, I can see why there's a four there. Four is the square root of 16, okay. So when you square it, you'll get two 16s, which factor out. So you're gonna get 16 times one minus sine squared theta. It's like, hmm, my trigonometric identity sense is tingling right now, right? One minus sine squared, that feels like it's the Pythagorean identity, right? Cosine squared theta plus sine squared theta equals one. If you subtract cosine from both sides, you get, excuse me, if we subtract sine from both sides, you're gonna get cosine squared theta equals one minus sine squared. Like so, it's like, oh, I can substitute one minus sine squared with cosine squared. I'm gonna put that in right there. So that's gonna give us the square root of 16 times cosine squared theta. So just like on the last example, you have a perfect square 16 and a perfect square cosine squared. So the square root is gonna be four times the absolute value of cosine theta. Which again, in calculus, because of other situations we're not considering right now, the absolute value generally can be dropped and you would just say four cosine theta at this situation. So in retrospect, you see exactly why this was a good substitution because of the Pythagorean relationship between sine and cosine. By substituting a sine in for x, you can turn this 16 minus x squared, this difference of squares. You can turn it into a perfect square by the Pythagorean identity and therefore you get the square root, which in the calculus setting is something you're not gonna want there.