# Right Angle Trigonometry

Problem was initially posted in a technical math section, so is missing the homework template.
From a position 150 ft above the ground, an observer in a building measures angles of depression of 12° and 34° to the top and bottom, respectively, of a smaller building, as in the picture on the right. Use this to find the height h of the smaller building.

I have found that the angle between the observer and the ground is 56°; however, when I use the tangent function to get the distance between the smaller building and the observer [150 tan(56)] I get the wrong answer and I do not understand why. The only solutions to this problem that I have found involve the law of sines, which has not been covered yet. If I knew the distance between the observer and the smaller building (let's call it x), then I would be able to construct a right triangle from the observing to above the smaller building, where the triangle would have acute angles 78 and 12 and a leg of length x; however, I would not know how to proceed from there.

My questions are: why can I not use [150 tan(56)] to calculate x, how do I calculate x, and how do I find h from the triangle with leg x and angles 78 and 12.

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Mark44
Mentor
From a position 150 ft above the ground, an observer in a building measures angles of depression of 12° and 34° to the top and bottom, respectively, of a smaller building, as in the picture on the right. Use this to find the height h of the smaller building.

I have found that the angle between the observer and the ground is 56°; however, when I use the tangent function to get the distance between the smaller building and the observer [150 tan(56)] I get the wrong answer and I do not understand why. The only solutions to this problem that I have found involve the law of sines, which has not been covered yet. If I knew the distance between the observer and the smaller building (let's call it x), then I would be able to construct a right triangle from the observing to above the smaller building, where the triangle would have acute angles 78 and 12 and a leg of length x; however, I would not know how to proceed from there.

My questions are: why can I not use [150 tan(56)] to calculate x, how do I calculate x, and how do I find h from the triangle with leg x and angles 78 and 12.
A drawing would be very helpful.

One thing I need to ask - is your calculator in degree mode?

FactChecker
I did my best to draw it, and yes, my calculator is in degree mode. I used variables to represent any quantity that wasn't explicitly given to me by the problem. By the complement rule, I think theta is 56°; however, I get the wrong answer for x when I do [150 tan(56)]. It is apparently solvable without the law of sines as the book has not covered that idea yet.

I believe that A is 90° and that B is 78°. I want to find the variable "h," which is the height of the small building.

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Mark44
Mentor
Your equation of x = 150 tan(56°) is fine. I did it another way, but the two ways are equivalent. All you need is another equation that involves x and h.

In your drawing, extend the line segment of length h all the way to the top so that you have a rectangle that is 150' by x'. Using the 34° you can use the dimensions of the triangle whose acute angle is 34° - that's your second equation. You don't need the Law of Sines or the Law of Cosines - just some ordinary right triangle trig.

BTW, this is a homework problem, so I am moving it to the Homework & Coursework sections, which is where problems of this sort should be posted.

Your equation of x = 150 tan(56°) is fine. I did it another way, but the two ways are equivalent. All you need is another equation that involves x and h.

In your drawing, extend the line segment of length h all the way to the top so that you have a rectangle that is 150' by x'. Using the 34° you can use the dimensions of the triangle whose acute angle is 34° - that's your second equation. You don't need the Law of Sines or the Law of Cosines - just some ordinary right triangle trig.

BTW, this is a homework problem, so I am moving it to the Homework & Coursework sections, which is where problems of this sort should be posted.
Oh, it looks like I forgot to subtract h from 150! The answer is then 103. Thank you for the advice!

Problem was initially posted in a technical math section, so is missing the homework template.
From a position 150 ft above the ground, an observer in a building measures angles of depression of 12° and 34° to the top and bottom, respectively, of a smaller building, as in the picture on the right. Use this to find the height h of the smaller building.

I have found that the angle between the observer and the ground is 56°; however, when I use the tangent function to get the distance between the smaller building and the observer [150 tan(56)] I get the wrong answer and I do not understand why. The only solutions to this problem that I have found involve the law of sines, which has not been covered yet. If I knew the distance between the observer and the smaller building (let's call it x), then I would be able to construct a right triangle from the observing to above the smaller building, where the triangle would have acute angles 78 and 12 and a leg of length x; however, I would not know how to proceed from there.

My questions are: why can I not use [150 tan(56)] to calculate x, how do I calculate x, and how do I find h from the triangle with leg x and angles 78 and 12.
wherw did you get 56 degree?

Mark44
Mentor
wherw did you get 56 degree?
Fill in the unknown angles based on the known angles.

FactChecker
Gold Member
Oh, it looks like I forgot to subtract h from 150! The answer is then 103. Thank you for the advice!
Good. In the future, it would be good to show the intermediate results of your calculations. Then it may be possible for you and others to spot a mistake quickly.

symbolipoint
Homework Helper
Gold Member
Good. In the future, it would be good to show the intermediate results of your calculations. Then it may be possible for you and others to spot a mistake quickly.
What? Post #6 comes six years after the initial few posts. Maybe the original poster member has moved on already.

PeroK
FactChecker
Gold Member
What? Post #6 comes six years after the initial few posts. Maybe the original poster member has moved on already.
Just trying to help. I didn't notice the dates of the posts.

symbolipoint
Homework Helper