Solving a Tower Height with Cosine Law

[uranium_235]

Question:
A man is looking at the top of a tower. The angle of elevation to the top of the tower is 10 degrees. 100m closer to the tower, a man has an angle of elevation to the top of the tower of 20 degrees, how tall is the tower?

My Problem:
I can solve this ver easily by recognizing that an isosceles triangle is formed when you draw the line of sight of the closer man, and since one of the sides of the isosceles is 100m the other side is as well, then I solve the smaller right triangle with tangents and get an answer approximately equal to 36m. But, this unit is on Cosine law, and I don't see how I can solve this with cosine law. Help?

 

[dextercioby]

Question:
A man is looking at the top of a tower. The angle of elevation to the top of the tower is 10 degrees. 100m closer to the tower, a man has an angle of elevation to the top of the tower of 20 degrees, how tall is the tower?

My Problem:
I can solve this ver easily by recognizing that an isosceles triangle is formed when you draw the line of sight of the closer man, and since one of the sides of the isosceles is 100m the other side is as well, then I solve the smaller right triangle with tangents and get an answer approximately equal to 36m. But, this unit is on Cosine law, and I don't see how I can solve this with cosine law. Help?

Yes you figured out well there should be an isosceles triangle somewhere.The important thing is to apply the "cosine law" in that isosceceles triangle to find the only side u don't know.That side is actually the hypotenuse in a rectangular triangle that has the tower as one of the sides.Then u just have to apply the definition of the function "sine" and u'll "home safe".

 

[PICsmith]

Hi uranium_235,
You're right, this prob is much easier to solve your way, however i got x = 100*sin(20) = 34.2, not quite 36. The law of cosines is good if you know 2 sides and 1 certain angle. You could apply this law by finding the length of the bottom (along the ground) of the right triangle (by the tower) and then: a^2 = b^2 + c^2 - 2bc*cos(A), where A = 20 degrees, b = length of bottom, and c = 100m.

 

[BobG]

Herr Schlauberger's right.

And the cosine law is a little bit of a convoluted route to take to the solution. A better question would have been how far the top of the tower was from the further man, a problem where the cosine law would be the most efficient solution.