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 feet away to be 30 degrees 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 feet away let's say the man is somewhere over here so this is 800 feet 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 sohcahtoa let's focus on the soul part of circator what it means is that sine theta is equal to the opposite side divided by the hypotenuse now the cop part means cosine theta is equal to the adjacent side divided by the hypotenuse and toa tangent theta is equal to the opposite side divided by the adjacent side so we have three trigonometric functions applicable to the situation we have the sine ratio the cosine and the tangent ratio now which one is most suitable to this specific problem should we use sine cosine or tangent so the angle that we have is 30 degrees opposite to the angle is the height of the building which we're looking for adjacent to the angle is 800 feet and the hypotenuse is across the 90 degree 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 sine or cosine we must use tangent so tangent theta as we mentioned before is equal to the opposite side divided by 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 divided by 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 times tangent of 30 degrees now tangent 30 is radical three radical three divided by three so the height of the building the exact answer is 800 root 3 divided by 3. if you want the decimal value oh by the way make sure your calculator is in degree mode 800 root 3 over 3 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 root 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 root 3 across the 90 is 2. perhaps you learned that in geometry so tangent 30 is equal to the opposite side which is one divided by the adjacent side relative to 30 which is root three now one over root three is the same as root three over 3. you need to rationalize the denominator so this becomes root 3 over 3. let's try this one calculate the angle of elevation measured from a point on the ground to a 50-foot tree that is 20 feet away from the tree now for all 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 divided by 20 opposite divided by the adjacent side 50 over 20 is the same as 5 over 2 you can cancel the zeros and 5 divided by 2 is 2.5 so tangent theta is 2.5 but how can we find theta to find the angle theta we need to take the inverse tangent of 2.5 so if you type an inverse tan 2.5 you should get an answer of 68.2 degrees so that's what you need to do whenever you're looking for an angle you can use the inverse tan function inverse sine 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 gonna draw just any type of shape and let's say this is the water and here is the boat so this is the horizontal line and here is 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 degrees so that's the angle of depression the height of the observation tower is 100 feet 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 degrees 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 degrees is equal to 100 and if we divide both sides by 10 10 degrees x is 100 divided by tangent of 10 degrees so what's tangent of 10 degrees tangent of 10 degrees is about 0.17633 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 feet 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'm going 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