In this video, we're going to focus on solving word problems. Typically, uh, angle of elevation and depression word problems. Let's start with this one. A man measures the angle of elevation between the ground and a building 800 ft away to be 30°. How tall is the building? So, let's say that the height of the man is irrelevant to the height of the building. So, here's the ground and let's say this is the building. With these problems, typically you need to draw a right triangle. So, let's draw the line of sight from the ground to the building. The angle of elevation is the angle above the horizontal. So, let's say this is the horizontal line. This is the angle of elevation. The angle of depression is the angle below the horizontal line. So 40 would represent the angle of depression. 30 would represent the angle of elevation. Now the building is 800 ft away. Let's say the man is somewhere over here. So this is 800 ft. And our goal is to solve for the height of the building. What equation can we use? Now you need to be familiar with the terms SOA. Let's focus on the so part of SOA. What it means is that sin theta is equal to the opposite side divided by the hypotenuse. Now the co part means cossine theta is equal to the adjacent side / the hypotenuse and toa tangent theta is equal to the opposite side / the adjacent side. So we have three trigonometric functions applicable to this situation. We have the sign ratio, the cosine and the tangent ratio. Now, which one is most suitable to this specific problem? Should we use s cosine or tangent? So, the angle that we have is 30°. Opposite to the angle is the height of the building, which we're looking for. Adjacent to the angle is 800 ft and hypotenuse is across the 90° angle. Since we're looking for the height of the building, which is the opposite side, and we have the adjacent side, we have to use the tangent ratio. We don't know anything about the hypotenuse of the triangle. So, we can't use S or cosine. We must use tangent. So, tangent theta as we mentioned before is equal to the opposite side / the adjacent side. And the angle is 30. So tangent 30 is equal to the opposite side which is h the height of the building / the adjacent side which is 800. So let's multiply both sides by 800. So these two will cancel. Therefore the height of the building is 800* tangent of 30°. Now tangent 30 is radle 3 radle 3 / 3. So the height of the building the exact answer is 800 3 / 3. If you want the decimal value, oh by the way, make sure your calculator is in degree mode. 800 3 over3 is about 461.88. So that's how tall the building is. By the way, for those of you who are wondering why the exact value of tangent 30 is 3 over 3, you can find this out using the 30 60 90 reference angle or reference triangle. Across the 30 is 1. Across the 60 is roo3 across the 90 is 2. Perhaps you learned that in geometry. So tangent 30 is equal to the opposite side which is 1 / the adjacent side relative to 30 which is roo3. Now 1 over3 is the same as roo3 over3. You need to rationalize the denominator. So this becomes roo3 over3. Let's try this one. Calculate the angle of elevation measured from a point on the ground to a 50ft tree that is 20 ft away from the tree. Now, for all of these problems, typically you're going to draw some sort of right triangle. So, let's say this is the point at which we're going to measure the angle of elevation. So, let's call it theta. So, this is going to be the tree, which The height of the tree, we know it to be 50 feet and it's 20 feet away from the tree. That's the point of interest. So, if we're given the height and the distance, how can we calculate the angle of elevation? What would you do to figure this out? So, again, we need to use the tangent ratio. tangent theta is 50 / 20 opposite / the adjacent side. 50 is the same as 5 / 2. You can cancel the zeros and 5 / 2 is 2.5. So tangent theta is 2.5. But how can we find theta? To find the angle theta, you need to take the inverse tangent of 2.5. So if you type in inverse tan 2.5, you should get an answer of 68.2°. So that's what you need to do whenever you're looking for an angle. You can use the inverse tan function, inverse sign, inverse cosine, depending on what two sides of the triangle that you have. Here's another one for you. A man on a 100 foot observation tower measures the angle of depression of a boat to be 10 degrees. How far is the boat from the tower? So let's say this is the tower. I'm just going to draw just any type of shape. And let's say this is the water. And here's the boat. So this is the horizontal line and here's the line of sight between the person in the tower and the boat. So here's the triangle that we want to draw. So the angle of depression is the angle that's below the horizontal line which is 10°. So that's the angle of depression. The height of the observation tower is 100 ft. Now we're going to assume that the height of the boat is negligible. Our goal is to find out how far is the boat from the tower which is basically x. So this side of the triangle is also x. Notice that we need to use the tangent ratio again. tangent of 10° is equal to the opposite side which is opposite to 10 is 100 divided by the adjacent side or the side next to 10 which is x. So if we multiply both sides by x, x tan 10° is equal to 100. And if we divide both sides by tan 10°, x is 100 / tangent of 10°. So what's tangent of 10°? tangent of 10° is about.17 633. So 100 divided by that number is about 567.1. So that's how far away the boat is from the shoreline or from the observation tower. it's 567 ft away from it. So to review, whenever you're dealing with angle of elevation problems, just remember the angle is above the horizontal line. And for angle depression or I meant to say angle of depression for those type of problems, the angle is below the horizontal line. So, make sure you remember that whenever you're solving these types of word problems. So, that is it for this video. Thanks for watching and have a great day.