Word problem question

Question

The CN Tower in Toronto, Canada, is the tallest free-standing structure in North America. A woman on the observation deck, 1150 ft above the ground, wants to determine the distance between two landmarks on the ground below. She observes that the angle formed by the lines of sight to these two landmarks is 𝜃 = 45°. She also observes that the angle between the vertical and the line of sight to one of the landmarks is a = 65° and that to the other landmark is b = 54°. Find the distance between the two landmarks. (Round your answer to the nearest foot.)

Answer 1

Let

T be the point at the top of the tower,

G be the point at the bottom of the tower,

A be the landmark with angle a = 65°,

B be the landmark with angle b = 54°.

We are are to calculate the distance AB, when given

the distance TG = 1150,

the angle ATG = a = 65°,

the angle BTG = b = 54°,

the angle ATB = θ = 45°.

1) By definition of cosine in the right angle triangle ATG

AT = TG/cos(a)

2) By definition of cosine in the right angle triangle BTG

BT = TG/cos(b)

3) By the Law of Cosines in the triangle ATB

AB² = AT² + BT² - 2×AT×BT×cos(θ)

= (TG/cos(a))² + (TG/cos(b))² - 2×TG²×cos(θ)/(cos(a)×cos(b))

= TG²×(1/cos²(a) + 1/cos²(b) - 2×cos(θ)/(cos(a)×cos(b)))

AB = TG×√(1/cos²(a) + 1/cos²(b) - 2×cos(θ)/(cos(a)×cos(b)))

Substituting actual numbers

AB = 1150×√(1/cos²(65°) + 1/cos²(54°) - 2×cos(45°)/(cos(65°)×cos(54°))) = 1924