Welcome back, everyone. So here we have an example where we need to use our understanding of right triangles and trigonometry to see if we can solve this story problem. So let's see what we've got. Here we have that a hiking path can be traced from a mountain lodge at an elevation of 6,500 feet to a scenic viewpoint in a canyon at an elevation of 4,300 feet. We have that the path spans 4,400 feet, and we're asked to determine the angle of inclination of the hiking path. Okay. Now to understand this, I actually drew this sketch out just to give us a general understanding of what's going on. And I will say that the sizes here are not necessarily drawn to scale, but this is going to give us a general understanding of where things are positioned. Now we have that this mountain lodge is at an elevation of 6,500 feet, which is going to be up here. So this distance would be 6,500 feet. Now we have that we're trying to reach a destination that is at an elevation of 4,300 feet, and that is going to be this distance down here.

And what we're trying to do is see what this is going to be, because we have that the path spans 4,400 feet. So that means that this path to get to our destination is 4,400 feet. And what we're trying to do is find the angle of inclination, and we can find this if we draw a straight line right here. Because this straight line is going to show us what the angle of inclination is, which is that angle. Now notice that we have a right triangle that forms when we do this.

But the height of this side of the triangle is not going to be the whole 6,500 feet; it's going to be some other height that we'll call h. And the reason why is because 6,500 feet is this entire distance, not just that distance. So what we need to do is figure out how we can use all this information to solve for this angle. Well, something that I see is we can actually figure out what this height is because notice that we have this distance as well as that distance. And what our height h is going to be is going to be the 6,500 feet minus the 4,300 feet, because that's going to give us this distance from the 4,300 feet of elevation to the mountain lodge. So 6,500 minus 4,300 will give us 2,200 feet. And this right here is the height that we are dealing with.

So now that we have the height, what we need to do is solve for the angle, and we can solve for this angle using SOHCAHTOA. SOHCAHTOA is this memory tool that we use to relate the trigonometric functions to the sides of the right triangle. Now what I noticed in this problem is that we have the opposite side of this triangle. So if we're looking at this angle, the opposite side is going to be 2,200 feet, and then the hypotenuse is 4,400 feet. And the trigonometric function that uses the opposite and the hypotenuse is the sine. So we can see that the sine of our angle theta is equal to the opposite side of this triangle, which is 2,200 feet divided by the hypotenuse, which is 4,400 feet. Now to solve for our angle theta, I'm going to take the inverse sine on both sides of the equation. That's when we get the sine and the inverse sine to cancel on the left side, leaving us with just theta.

And theta is going to equal the inverse sine of 2,200 / 4,400. Now 2,200 over 4,400 is something that can reduce. This is equivalent to the inverse sine of 1 half, because 2,200 goes into 4,400 two times, so this would reduce to a half. And the inverse sine of 1 half comes out to an angle of 30 degrees. So 30 degrees is our angle of inclination and the solution to this problem. So that's how you can solve this. Hope you found this video helpful. Thanks for watching.