Height of Radio Tower: Using the Law of Sines

  • Thread starter StarkyDee
  • Start date
  • Tags
    Trig
In summary, two boys measured the angle of elevation of a nearby radio tower from two different locations and also measured the height of a window. Using the law of sines, they were able to calculate the height of the tower to be approximately 55 feet.
  • #1
StarkyDee
2 boys desiring to estimate the height of a nearby radio tower measured the angle of elevation at their house and found it to be 52 degrees. they took a second measurement from the second-story window and found the angle of elevation to be 44 degrees. they next measured the window to be 24 feet above the ground. to the nearest foot, what is the height of the radio tower?
Law of sines:::

i can't figure out how to use new latex for this ...
sin a / a = sin b / b = sin c / c

thanks
 
Physics news on Phys.org
  • #2
A diagram of what you mean would really really help.
 
  • #3
I'm pathetic with Latex as well, so bear with me.

2 boys desiring to estimate the height of a nearby radio tower measured the angle of elevation at their house and found it to be 52 degrees. they took a second measurement from the second-story window and found the angle of elevation to be 44 degrees. they next measured the window to be 24 feet above the ground. to the nearest foot, what is the height of the radio tower?

Okay, call the height of the tower 'a' and the distance along the ground toward the tower 'c'. You need to think of a right-angled triangle with base 'c' and opposite leg 'a'.

When the boys take their measurements from the second-floor window, think of a different triangle, this one with opposite side (a - 24)ft in height. You may notice that the base 'c' does not change between these two locations.

Therefore

a/(sin 52) = c/(sin (90 - 52)) = c/(sin 38)

and

(a - 24)/(sin 44) = c/(sin(90 - 44)) = c/(sin 46)

Since

a/(sin 52) = c/(sin 38)

a = c(sin 52)/(sin 38)

And also, since

(a - 24)/(sin 44) = c/(sin 46)

a = c(sin 44)/(sin 46) + 24

So, now we know that

c(sin 44)/(sin 46) + 24 = c(sin 52)/(sin 38)

And that

c = 24/[(sin 52)/(sin 38) - (sin 44)/(sin 46)]

From before, we found that

a = c(sin 52)/(sin 38)

So substitute your value for c into this equation and that will yield your answer.
 
  • #4
AD- thanks for the calculations. but i still don't understand how to diagram the second triangle? the first triangle is a right triangle,
i think i have the first triangle right..but where does the 2nd one go?
 
  • #5
I've drawn a diagram in paint, but it is not attaching because it is apparently too big, despite the fact that I've gotten the file size down to 9KB. Perhaps you could PM me your e-mail address and I can send it to you that way.
 
  • #6
Did you get the e-mail, Starky? Or did it not go through?
 
  • #7
oh ok! thanks so much for the graph, now i understand what your talking about with that diagram. i can't believe i couldn't figure that out- it's so easy now that i look at your picture- thanks again andrew.
sincerely,
~david
 

FAQ: Height of Radio Tower: Using the Law of Sines

What is the Law of Sines?

The Law of Sines is a trigonometric law that relates the lengths of the sides of a triangle to the sine of its angles. It states that the ratio of the length of a side to the sine of the opposite angle is equal for all sides and angles in a triangle.

How do I use the Law of Sines to solve a triangle?

To use the Law of Sines, you need to have at least one pair of a side and its opposite angle. Then, you can use the law to find the missing sides or angles by setting up and solving a proportion. You may also need to use the Law of Cosines to find a missing side or angle if you do not have enough information.

Can the Law of Sines be used on any triangle?

Yes, the Law of Sines can be used on any triangle, regardless of its shape or size. However, it is most commonly used on oblique triangles, which have no right angles.

What is an ambiguous case in the Law of Sines?

An ambiguous case in the Law of Sines occurs when there are multiple possible solutions for a triangle. This happens when you have two sides and an angle opposite one of them, and the sine of that angle is greater than one. In this case, there are two possible triangles that could be formed with the given information.

Are there any limitations to the Law of Sines?

Yes, the Law of Sines has a few limitations. It can only be used to solve triangles, and it requires at least one known side and its opposite angle. It also cannot be used if the given angle is a right angle, as the sine of a right angle is undefined. In addition, the Law of Sines can produce ambiguous solutions in certain cases, as mentioned in the previous question.

Back
Top