Problem 52 - Precalculus Mathematics for Calculus

Question

The CN Tower in Toronto, Canada, is the tallest free-standing structure in North America. A woman on the observation deck, \(1150 \mathrm{ft}\) 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 \(43^{\circ} .\) She also observes that the angle between the vertical and the line of sight to one of the landmarks is \(62^{\circ}\) and to the other landmark is \(54^{\circ} .\) Find the distance between the two landmarks.

Step-by-Step Solution

Answer

The distance between the two landmarks is approximately 1333.23 feet.

Step 1: Identify Known Angles in the Triangle

The woman observes the angle between her lines of sight to the two landmarks is \(43^\circ\). This is the angle included in the triangle formed by her lines of sight and the direct paths to each landmark. We denote the angle between the vertical and the line of sight to the first landmark as \(62^\circ\) and the second landmark as \(54^\circ\).

Step 2: Calculate the Remaining Angles

In the triangle formed by the observation deck and the landmarks, we need to calculate the angles that the landmarks create with the vertical. The angle for the first landmark is the complement of \(62^{\circ}\) with respect to \(90^{\circ}\), which calculates to:\[90^\circ - 62^\circ = 28^\circ. \]Similarly, for the second landmark, the angle with the vertical is:\[90^\circ - 54^\circ = 36^\circ.\]

Step 3: Use the Law of Sines

The Law of Sines relates the sides of a triangle to the sines of its angles. Applying it here will help find the distance between the landmarks:The angles known in the triangle from the observation deck are \(28^\circ\), \(36^\circ\), and \(43^\circ\). The side opposite \(43^\circ\) is the distance we need to find. For the side opposite \(28^\circ\), its length is the height of the tower, \(1150\) ft.According to the Law of Sines:\[\frac{d}{\sin(43^\circ)} = \frac{1150}{\sin(36^\circ)},\]where \(d\) is the distance between the two landmarks.

Step 4: Solve for the Distance

We rearrange the equation from Step 3 to solve for \(d\):\[d = 1150 \times \frac{\sin(43^\circ)}{\sin(36^\circ)}.\]Using a calculator, find \(\sin(43^\circ) \approx 0.6820\) and \(\sin(36^\circ) \approx 0.5878\).Substitute these values into the equation:\[d \approx 1150 \times \frac{0.6820}{0.5878} \approx 1333.23 \text{ ft}.\]

Key Concepts

Law of Sines

The Law of Sines is a helpful trigonometric tool used to solve for unknown sides or angles in a triangle. It is particularly useful in cases involving oblique triangles, which are non-right angled triangles. The law states that the ratio of the length of a side to the sine of its opposite angle is constant for all sides of the triangle. It can be expressed mathematically as:
\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]where \(a\), \(b\), and \(c\) are the sides of the triangle, and \(A\), \(B\), and \(C\) are the respective opposite angles. In our exercise, the challenge was to find the distance between two landmarks using this relationship. Once the angles and one side (the height of the CN Tower) are identified, the Law of Sines enables us to set up an equation to calculate the unknown distance.

Angle of Elevation

The angle of elevation is an angle formed between the horizontal line of sight and the line of sight up to an object. In trigonometry problems like this exercise, angles of elevation are key in determining distances in a triangle.
The woman on the observation deck measures the angles her lines of sight make with the vertical. These measured angles relate to angles of elevation, as they help calculate the total angles within the triangle formed by the observation deck and the landmarks below.
In the exercise, the angle of elevation to one landmark was \(62^{\circ}\), and to the other was \(54^{\circ}\). By calculating the complementary angles with the vertical, we establish two angles of \(28^{\circ}\) and \(36^{\circ}\) respectively, which are then used in the Law of Sines.

Distance Calculation

The final goal in our exercise was to determine the distance between two landmarks using trigonometric principles. With the Law of Sines and calculated angles, we set up an equation to solve for the unknown distance.

• We identified that the height of the CN Tower, \(1150\) ft, is one side of the triangle, opposite to the \(28^{\circ}\) angle.
• The angle \(43^{\circ}\) between the lines of sight is the included angle between the primary directions to the landmarks.

By applying the Law of Sines as \[\frac{d}{\sin(43^{\circ})} = \frac{1150}{\sin(36^{\circ})}\]we calculate the approximate distance \(d\). This involves manipulating the formula to find:

• \(d = 1150 \times \frac{\sin(43^{\circ})}{\sin(36^{\circ})}\)

Using approximate sine values of \(\sin(43^{\circ}) \approx 0.6820\) and \(\sin(36^{\circ}) \approx 0.5878\), the calculation provides the final distance between the landmarks as \(1333.23\) ft.