# Question Video: Using the Sine Rule to Calculate an Unknown Length of a Triangle in a Real-Life Problem Mathematics

Cities π΄, π΅, and πΆ are located such that city π΄ is west of city π΅, city πΆ is on a bearing of 35Β° from city π΅, and city πΆ is 100 miles from city π΄ and 70 miles from city π΅. Find the distance between cities π΄ and π΅ giving your answer to one decimal place.

04:16

### Video Transcript

Cities π΄, π΅, and πΆ are located such that city π΄ is west of city π΅, city πΆ is on a bearing of 35 degrees from city π΅, and city πΆ is 100 miles from city π΄ and 70 miles from city π΅. Find the distance between cities π΄ and π΅ giving your answer to one decimal place.

Letβs begin by sketching a diagram. Remember this doesnβ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. Weβll start by adding city π΅ and a north line to our diagram. City π΄ is some distance west of city π΅. That means that the angle between the north line and the line π΄π΅ must be 90 degrees.

City πΆ is on a bearing of 35 degrees from city π΅. Remember bearings are measured in a clockwise direction from the north line. So the angle between the north line and π΅πΆ is 35 degrees. We know that city πΆ is 100 miles from city π΄ and 70 miles from city π΅. We can calculate the measure of the angle at π΅ by adding together 90 and 35. 90 plus 35 is 125 degrees. So the measure of the angle at π΅ is 125.

Weβre trying to calculate the distance between city π΄ and π΅. Weβll call that side lowercase π because it sits directly opposite the angle πΆ. In order to use the law of sines to calculate the length of the side π, weβd first need to know the measure of the angle at πΆ.

What we can do first though is calculate the measure of the angle at π΄ since we know the length of the side π. Letβs use the law of sines then to calculate the measure of this angle. We can use either of these versions. But since weβre first trying to find the missing length, itβs sensible to use the first one. This will minimise the amount of rearranging we need to do.

Since weβre trying to calculate the measure of the angle at π΄ and we know the measure of the angle at π΅, weβll use the first two parts of the formula: sin π΄ over π equals sin π΅ over π. Substituting what we know into this formula gives us sin π over 70 equals sin 125 over 100.

We can solve this equation by multiplying both sides by 70 to get sine of π equals 70 sine of 125 all over 100. That gives us a value of 0.573. To calculate the value of π then, weβll find the inverse sine of both sides of this equation. π is equal to the inverse sine of 0.573 which is 34.988 degrees. We wonβt round this answer just yet; instead, weβ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 πΆ by subtracting the two known angles from 180. The measure of the angle at πΆ then is 20.011 degrees. And now that we know the measure of the angle at πΆ, we can use the sine rule once again to find the length labelled π.

Remember since weβre finding a side, we can now use the second version for the law of sines. Weβll use π over sin π΅ equals π over sin πΆ. Substituting what we know into this formula gives us 100 over sine of 125 equals π over sine of 20.011.

Weβll solve this equation by multiplying both sides by sine of 20.011. π 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 π΄ and π΅ is 41.8 miles.