Question

The distance between two buildings is labeled in red below. You can see the closest edge of each building with an angle of view of 65°
. Your distance to one edge is 22m and to the other is 17 m. To find the distance between the two buildings (red), will you use Law of Sines or Law of Cosines? What is the distance?

Answer 1

To determine the distance between the two buildings using the information provided, we can indeed use the Law of Sines or the Law of Cosines. However, we need to set up the problem properly first.

1. **Understanding the setup:**
- Let point \( A \) be your position, point \( B \) be the closest edge of the first building, and point \( C \) be the closest edge of the second building.
- From point \( A \):
- The distance to edge \( B \) is \( AB = 22 \) m.
- The distance to edge \( C \) is \( AC = 17 \) m.
- The angle \( BAC = 65^\circ \) (the angle subtended between lines \( AB \) and \( AC \)).

2. **Choosing the Law of Sines or Cosines:**
Because you know two sides and the included angle, the Law of Cosines is appropriate to find the distance \( BC \) between the two buildings directly.

3. **Using the Law of Cosines:**
The Law of Cosines states:
\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]
where:
- \( a = AC = 17 \, m \)
- \( b = AB = 22 \, m \)
- \( C = 65^\circ \)
- \( c = BC \) (the distance we want to find)

Substituting the known values:
\[
BC^2 = 17^2 + 22^2 - 2 \cdot 17 \cdot 22 \cdot \cos(65^\circ)
\]

4. **Calculating:**
\[
BC^2 = 289 + 484 - 2 \cdot 17 \cdot 22 \cdot \cos(65^\circ)
\]
\[
BC^2 = 773 - 748 \cdot \cos(65^\circ)
\]
Using \( \cos(65^\circ) \approx 0.4226 \):
\[
BC^2 = 773 - 748 \cdot 0.4226 \approx 773 - 316.67 \approx 456.33
\]
\[
BC \approx \sqrt{456.33} \approx 21.4 \, m
\]

So, the distance between the two buildings is approximately **21.4 meters**.