Quickly master how to solve Trigonometric functions with Double Angle Identities with easy tutorial. Watch more lessons like this and try our practice at [ Ссылка ]
In this lesson, we will learn how to make use of the double-angle identities, a.k.a. double-angle formulas to find the sine and cosine of a double angle. It’s hard to simplify complex trigonometric functions without these formulas.
let's talk about double angle identity. Double angle, what is double angle? Here it is. Let's take a look at the trigonometry identity chart, right here. Double angle identity, double angle, that means what? Look at the angle theta, double that two theta. So it's sine two theta, we have this formula. cosine two theta, we have these formulas. So guys, cosine two theta is actually the tricky one. Because when you see a question, you don't know when you see cosine two theta, you don't know whether to use the first one, the second one or the third expression, right? And then, you know what, don't worry. I'll teach you the trick, okay? I'll teach you the trick so when you see a question, and it depends on how the question's laid out, you'll be able to know whether to use the first expression, second expression or third expression for cosine two theta, okay? And now here, tangent two theta equals this expression. So in a way, sine and tangent double angle, they are very easy, right? They are very easy because there's only one formula for them. But for cosine there are three formulas for that, okay?
So don't worry, later on we'll learn all the tricks you need to know, okay? Good. Now, coming back to this question. Here we have a number, sine, cosine. If you look at the trigonometry identity chart, a number, sine, cosine, there's nowhere else where you see that. A number followed by sine cosine, you don't see that anywhere else, see? Not in quotient identities, not in reciprocal identities, not in Pythagorean identities.
Not in sum and difference identities, right? So, you only see that in what? In double angle identities for sine, see? A number followed by sine and cosine, right?
So, we know we want to use this formula for this particular question, sine two theta, right? So coming back to the question, and guys, here's the trick. Let's take out, now we know which formula we're gonna be using, right? We're gonna be using this formula. Sine two theta equals two sine theta, cosine theta, right? So, we know we're gonna be using this formula for this question, and guys, our approach here is we're gonna modify the formula we are given to something we want, okay? Check this out. It's actually really fun, okay? Look. Guys, here we have a sine, cosine, and here we have a sine, and cosine. That's nice, right? In fact, is because of that we know we have to use this formula. But now, look. The number in front is 14, now here we only have two, though. So you know what we do? Multiply the whole expression by seven. Now, so multiplying this in we get, seven sine two theta equals, and multiplying this into the right side of the equation, so seven times two we have 14, right? Sine theta, cosine theta, right? So basically is multiplying seven to the given formula, sine two theta, right?
So guys, you see we're getting one step closer because now check this out. 14, 14. sine, cos, sine, cos. We're getting actually very close. And the next thing we have to do is what? Let's replace, let's replace what? Let's replace theta, theta, by 6x, right? So, let's theta equals 6x. See here, replace theta by 6x, replace theta by 6x. Replace the angle theta here, by 6x. And by the way, there should be a bracket here. Okay, so what do we come up with? At the end, this is what we have, right? Check this out. We have 14 sine, cosine. Seven sine, two multiplied by theta. Okay, check this out. So we've got to replace theta by 6x, so here put 6x. Replace theta here, by 6x, right? And guys, same here, replace theta here by 6x. So guys, now look at the right side of the equation, read it out now. Fourteen, 14. Sine, sine. 6x. And cosine 6x. Guys, isn't this exactly what the question is trying to ask us? Fourteen sine 6x cos 6x, 14 sine 6x cosine 6x. Guess what? So, this thing actually equals what? Equals seven sine two times 6x, we get 12x. That's it. Done.
Right? How cool is that? So that's it. At the end, we know that 14 sin 6x cos 6x. Right here, equals seven sine 12x, equals seven sine 12x, right? How cool is that? Okay? Good.
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