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 è = 43°. She also observes that the angle between the vertical and the line of sight to one of the landmarks is a = 62° and that to the other landmark is b = 53°. Find the distance between the two landmarks.

I am having the same problem Steve had, when this same question was posted 2 years ago

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Notice , the wording is identical, just the angles have been changed.

is there a way to upload a picture?

No, but there are only a few points. Describe the picture.

If a and b are on the same side of the vertical line down from the tower, then a-b=9°. If not, a+b=115°

Where does the 43° angle come into play?

To find the distance between the two landmarks, we can make use of trigonometry and the given angles.

Let's denote the distance between the CN Tower and the first landmark as x, and the distance between the CN Tower and the second landmark as y.

Using trigonometry, we can set up a proportion based on the angles and the distances:

In the triangle formed by the CN Tower, the first landmark, and the vertical line, we have the angle a = 62°. Therefore, we can write:

tan(a) = x / 1150 ft

This gives us the equation:
x = 1150 ft * tan(a)

Similarly, in the triangle formed by the CN Tower, the second landmark, and the vertical line, we have the angle b = 53°. Therefore, we can write:

tan(b) = y / 1150 ft

This gives us the equation:
y = 1150 ft * tan(b)

Now, we know that the angle formed by the lines of sight to the two landmarks is è = 43°. This means that the angle between the lines connecting the CN Tower to the two landmarks is 180° - è = 180° - 43° = 137°.

In a triangle formed by the two landmarks and the connecting line, we have an angle of 90° at the CN Tower and the angle 137° opposite to the distance between the two landmarks. Using the law of sines, we can write:

sin(137°) / x = sin(90°) / d

where d is the distance between the two landmarks.

Rearranging the equation, we get:

d = x * sin(90°) / sin(137°)

Substituting the value of x that we obtained earlier, we have:

d = (1150 ft * tan(a)) * sin(90°) / sin(137°)

Now, we can substitute the values and calculate the distance:

a = 62°
b = 53°
è = 43°

Using a scientific calculator or a trigonometric table, we can find the values of sin(90°), sin(137°), tan(a), and tan(b). Plugging in these values, we can calculate the distance d.

Keep in mind to use the appropriate unit conversions for the distances.

Therefore, with these steps, we can determine the distance between the two landmarks using the given information and trigonometric principles.