Welcome back, everyone. So let's try this example. In this example, we are told the grand lighthouse on a coastal cliff stands 288 meters tall and is positioned approximately 2.3 kilometers inland from the shore of the sea. A seafarer on a sailboat directly in front of the lighthouse observes the top of the structure and records the angle of elevation as 3.4 degrees. Determine the distance in kilometers of the sailboat from the coastline to 2 decimal places. Okay. So this is a story problem. And what we're going to do is see if we can use our understanding of right triangles and trigonometry to solve this problem. So what I did is drew a diagram of this situation, and I'll say nothing's, like, specifically drawn to scale here, but this is the situation, basically. And what we have is this boat, which is measuring to the top of this lighthouse. And we're told that the lighthouse is 288 meters tall, so that is going to be the height of this lighthouse.

Now, what we also have in this problem is we have this sailboat, and the sailboat over here is measuring a distance from sailboat to the top of the tower. And it says that this angle of elevation is 3.4 degrees. Now what we also have is that this lighthouse is approximately 2.3 kilometers inland from the shore of the sea. So this is where the lighthouse is, and then the shore of the sea is right about over there. And then this distance is going to be 2.3 kilometers. And keep in mind, there are 1000 meters in a single kilometer. So 2.3 kilometers is the same thing as 2300 meters. So this is the distance there. And what we're trying to do is actually find this distance, which is the distance from the shore to the boat. So we'll call this distance $d$. Now to go about solving this problem, what we need to do is just think about how we can relate all of these sides. And if you look closely, you can see that we actually have a right triangle which forms because this is a right triangle that we're looking at, so I can use SOHCAHTOA.

Now what I notice is that the hypotenuse is not a value that is given to us. So using the sine and the cosine might not be the best idea. But I noticed that we have the opposite side of this triangle, as well as the adjacent side, so the tangent is our best way to go. So we have that the tangent of our angle theta is going to equal the opposite side of this triangle divided by the adjacent side. And what I can see from this triangle is that if we go opposite to the angle that we have, it's 288 meters. So we're going to have that the tangent is equal to 288 meters divided by, and then we have the adjacent side, which is this whole side of the triangle. So it's going to be 2300 plus our distance $d$, and then this tangent of our angle is 3.4 degrees. So this is going to be the angle within the tangent operation. And now from here, all I need to do is use some algebra to get this distance by itself, and I'm going to go ahead and do this over here.

So what I'm first going to do is take both sides of this equation. I'm going to multiply it by 2300 plus $d$. So I'm going to apply the left and the right side by this quantity, and what that is going to allow me to do is cancel these numbers on the right side of the equation. So all we're going to be left with is 288. And then on the left side, we're going to have 2300 plus $d$ times the tangent of 3.4 degrees. What I can actually do is take this tangent, and I can distribute it into these parentheses. We're going to have 2300 times the tangent of 3.4 degrees, and this is going to be plus $d$ times the tangent of 3.4 degrees. Now what I'm going to do from here is 2300 times the tangent of 3.4. I'm going to put that into my calculator to simplify this and make this equation a little bit easier.

So putting this into a calculator, this number comes out to about 136.65. This is going to be plus $d$ times the tangent of 3.4, and this is all going to be equal to 288. Now what I can do at this point is take this 136.65 and subtract it on both sides of the equation. That is going to get these to cancel on the left side. And then on the right side, we'll have 288 minus this number. Subtracting these 2 will give you 151.35, and that is all going to be equal to $d$ times the tangent of 3.4. Now at this point, all I need to do is get $d$ by itself, and I can do that by dividing both sides of this equation by the tangent of 3.4. And doing this will allow me to cancel the tangent of 3.4 on the left side of the equation, leaving us with just $d$. And then on the right side of the equation, we'll have 151.35 divided by the tangent of 3.4. Doing this should give you a value of approximately 2547.6 meters. And keep in mind that we are asked for the answer in kilometers in this problem. So what we need to do is convert this to kilometers by dividing this by 1000. So there's 1000 meters in a single kilometer. So our distance is going to be this number if we move the decimal place back 3 points, it's going to be about 2.55 kilometers. And this right here is the distance from the sailboat to the shore. So 2.55 kilometers is the distance, and that is the answer to this problem. So hope you found this video helpful. Thanks for watching, and please let me know if you have any questions.