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Trigonometric and Exponential Functions - Example 7

In mathematics, trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. Trigonometry is also the foundation of surveying.

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all right. Everybody's favorite game solving trig and metric equations. And now one thing to kind of notice is that we have to put this restriction here. That data lies between zero and two pi. Now, it might have been something else, but if we don't put it, restriction triggered. A metric equations like this actually have infinitely many solutions because they're what's called periodic. So if I just keep adding to pi to my solution, I'm just going to get a new, unique solution. But that's why we're truncating the angles to be just in one rotation as we go around the unit circle from zero all the way to to buy where we started. Okay, So if we want toe solve this equation, we just want to isolate data. And so notice that when we write sine squared data like this, what this really means is it sign of data all square and now we just kind of want to use order of operations to reverse this process. So this is something squared. So let's take the square root now. I'm taking the square root of something squared. This is very important to remember, and you're seeing why this is important because we're going to see things like this a lot. This is not signed data. It's actually absolute value of science data. So I have something squared and I take the square root. It's just absolute value. That's something. The square root of 3/4 is really just the square root of 3/4, and I can write that is the square to three over the square root of two, using properties of exponential. Sorry over the square to four, which is to since it's just square to three, divided by square to four, but square to force to. And so now remember, with absolute value, I have to remember think about two cases. Either sine theta is squared three over to or negative sign data is squared three over to Okay, so then we just got to go back to the unit. Circle the common values. So when is sign equal to square toe three over to or when it's signed, data equal to negative square to three over to Well, if I just look at my table of values, it's pretty easy. You have data can be pi over three. It's gonna give me Route three over to We can have 25 or three. That's also going to give me route three over to or on the lower half of the unit circle. I have four pi over three that's going to give me a Y value of negative Route three over to and then finally, we have 55 or three. That's also going to give me a value of negative. You're in three over to So we actually have four solutions to this equation. And now notice that all of these solutions come from just going around the circle one time. In other words, these angles air between zero and two. Fine. So here we have another equation involving coastline data, and we're trying to solve data between zero and two pi. So the goal is to isolate data or, in other words, isolate coastline data. But here we actually have a problem because this is cosine tooth data and this is cosine theta. We really don't have a way to combine does two quantities. But this is what we can dio. We can actually use a trick identity. So cosign tooth data is really the same thing as cosine of theater plus data. But if I used the cosign, some formula, the angle edition formula, this is coastline Square data minus sine squared data. So really, this equation it's cousin square data minus sine squared. Data plus goes on. Data equals zero. But I can go a step further. I can write sine squared data. Well, really negative sine squared data as follows one. See cosine squared. Data minus one. And this is coming from the Pythagorean identity. I'm just rearranging this equation. Sine squared data plus coastline squared data equals one. I just subtracted sine squared and then subtracted one so I can replace negative sine squared data with cosign squared data minus one. So I have coastline square data. Plus, because I absorb this minus cosine squared data minus one plus because I'm data or, in other words, to co sign squared Data plus co. Santa minus one equals zero. Okay, so why did I do all that work? Is this really so much better then when I started with Well, yes, because let's say that why is equal to cosign data or let's let's do X. Yeah, race that. So we'll have excess coastline data. I could have done why, But I just wanted to use X. So this is really, like two X squared plus X minus one. But this is a quadratic equation, and actually in factors, this is two X minus one and then X plus one. You can verify that when I foiled cell and get the same thing. Okay, but that means that to cosign data. If I plug back in cosign for X minus one times co signed data plus one equals zero. So I have a product of two factors equaling zero. I can use the zero product property and conclude that either to cosine theta minus one is zero or co sign data plus one zero. But this just means cosign data is one half, and this just means cosign data is negative one. So I'm looking for values of data between zero and two pi, for which cosign data is either one half or negative one. But we could just look at the unit circle, enlist those out. We have pi over three that's going to give me one half. We have pie that's going to give me negative one, and then finally, we also have five pi over three. That's again going to give me an X coordinate of what happened. So here are my solutions to this equation between zero and two by