Cities 𝐴, 𝐵, and 𝐶 are located such that city 𝐴 is west of city 𝐵, city 𝐶 is on a bearing of 35 degrees from city 𝐵, and city 𝐶 is 100 miles from city 𝐴 and 70 miles from city 𝐵. Find the distance between cities 𝐴 and 𝐵 giving your answer to one decimal place.
Let’s begin by sketching a diagram. Remember this doesn’t need to be to scale. However, it should be roughly in proportion so that we can check the suitability of any answers we get. We’ll start by adding city 𝐵 and a north line to our diagram. City 𝐴 is some distance west of city 𝐵. That means that the angle between the north line and the line 𝐴𝐵 must be 90 degrees.
City 𝐶 is on a bearing of 35 degrees from city 𝐵. Remember bearings are measured in a clockwise direction from the north line. So the angle between the north line and 𝐵𝐶 is 35 degrees. We know that city 𝐶 is 100 miles from city 𝐴 and 70 miles from city 𝐵. We can calculate the measure of the angle at 𝐵 by adding together 90 and 35. 90 plus 35 is 125 degrees. So the measure of the angle at 𝐵 is 125.
We’re trying to calculate the distance between city 𝐴 and 𝐵. We’ll call that side lowercase 𝑐 because it sits directly opposite the angle 𝐶. In order to use the law of sines to calculate the length of the side 𝑐, we’d first need to know the measure of the angle at 𝐶.
What we can do first though is calculate the measure of the angle at 𝐴 since we know the length of the side 𝑎. Let’s use the law of sines then to calculate the measure of this angle. We can use either of these versions. But since we’re first trying to find the missing length, it’s sensible to use the first one. This will minimise the amount of rearranging we need to do.
Since we’re trying to calculate the measure of the angle at 𝐴 and we know the measure of the angle at 𝐵, we’ll use the first two parts of the formula: sin 𝐴 over 𝑎 equals sin 𝐵 over 𝑏. Substituting what we know into this formula gives us sin 𝜃 over 70 equals sin 125 over 100.
We can solve this equation by multiplying both sides by 70 to get sine of 𝜃 equals 70 sine of 125 all over 100. That gives us a value of 0.573. To calculate the value of 𝜃 then, we’ll find the inverse sine of both sides of this equation. 𝜃 is equal to the inverse sine of 0.573 which is 34.988 degrees. We won’t round this answer just yet; instead, we’ll use its exact form in our future calculations.
Since we know that angles in a triangle add to 180 degrees, we can calculate the measure of the angle at 𝐶 by subtracting the two known angles from 180. The measure of the angle at 𝐶 then is 20.011 degrees. And now that we know the measure of the angle at 𝐶, we can use the sine rule once again to find the length labelled 𝑐.
Remember since we’re finding a side, we can now use the second version for the law of sines. We’ll use 𝑏 over sin 𝐵 equals 𝑐 over sin 𝐶. Substituting what we know into this formula gives us 100 over sine of 125 equals 𝑐 over sine of 20.011.
We’ll solve this equation by multiplying both sides by sine of 20.011. 𝑐 is equal to 100 over sine of 125 multiplied by sine of 20.011 which is 41.776.
Correct to one decimal place, the distance between cities 𝐴 and 𝐵 is 41.8 miles.