Hey everyone! In this problem, we're asked to find all solutions to the equation 4 times the cosine of 2 theta plus pi plus 8 is equal to 12. Now, this might look a little bit complicated at first, but we're still going to work through this the same exact way using our steps. So, of course, starting with step number 1, we want to go ahead and isolate our trigonometric function. In this case, our trigonometric function looks a little bit different. It is the full cosine of 2 theta plus pi. That's what we want to isolate. Now we can start by subtracting 8 from both sides, canceling over here, and leaving me with 4 times the cosine of 2 theta plus pi is equal to 12 minus 8, which is positive 4. Now from here, I can isolate this by dividing both sides by 4, canceling on that left side, leaving me just with my trigonometric function, cosine of 2 theta plus pi is equal to 4 divided by 4, which is positive 1. Now I've completed step number 1. My trigonometric function is isolated. It's all by itself. And I can move on to step number 2, finding all of my solutions on the unit circle.

This might seem a little bit strange to you at first because we're not just dealing with theta. We're dealing with 2 theta plus pi. But it really doesn't matter at this point in our process. It's not going to matter until later on in our steps because we're still trying to find a value for which the cosine is equal to 1. So, 2 theta plus pi, what does that need to be in order to get a cosine of 1? We're working through this the same way. So, coming to our unit circle here, where is my cosine going to be equal to 1? Well, I know that for 0, my cosine is 1, and that's actually the only angle for which that is true. So in order to get a cosine value of 1, that 2 theta plus pi needs to be equal to 0. So from here, we have completed step number 2. We found our solutions that are on our unit circle.

Now from here, if our domain is not restricted, we want to add \(2\pi n\) to each solution. And because we're asked to find all of our solutions and not given a restriction, we do need to go ahead and add \(2\pi n\) over here. So here I'm left with 0 + \(2\pi n\). Now, I can actually simplify that because that 0 isn't doing anything. So this is \(2\pi n\) on that right side. And then I'm left with 2 theta plus pi is equal to \(2\pi n\). That represents all of my solutions. But we have one final step here because we've completed step number 3, but step number 4 is to isolate theta. That's where we're actually going to deal with this 2 theta plus pi.

So from here, we're going to just go ahead and use algebra to isolate theta. In order to do that, I want this theta by itself. The first thing I want to do is subtract pi from both sides. Now it will cancel over here, leaving me with 2 theta is equal to \(2\pi n - \pi\). I can't really simplify that much. So I can go ahead and fully isolate theta by dividing both sides by 2. Now, that will cancel over here, leaving me with theta is equal to \(2\pi n - \pi\) divided by 2. Now we can simplify this a little bit further. This is technically correct because theta is by itself, but we can further simplify this because \(2\pi n\) divided by 2 will just leave me with \(\pi n\), and then I'm subtracting a \(\pi\) over 2. So my final answer is that theta is equal to \(\pi n - \frac{\pi}{2}\). And that represents all of my solutions to my original equation. Now even though this can get a little bit complicated, you're always just going to follow your steps and you will get to the right answer. Thanks for watching and let me know if you have any questions.