### Video Transcript

James, Anthony, and Jennifer stand at three points, π΄, π΅, and πΆ, respectively. Suppose that the measure of the angle π΄π΅πΆ is equal to 48 degrees, the measure of
the angle π΅π΄πΆ is equal to 54 degrees, and James is exactly 12 feet away from
Anthony. Find the distance between Anthony and Jennifer, to two decimal places.

When you present it with a lot of information, it can be really useful to begin by
sketching a diagram. Remember this diagram doesnβt need to be to scale. But it should be roughly in proportion so we can spot any mistake should they
occur.

In this case, we have a non-right-angled triangle, for which we know the measure of
two of the angles and the length of one of the sides given. Weβre looking to find the length of a second side. Thatβs the distance between the points π΅ and πΆ on our diagram. This means weβll need to use the law of sines.

We know that we canβt use the law of cosines to answer this problem since the law of
cosines requires at least two known sides and we know only the length of one. The sine rule says π over sin π΄ is equal to π over sin π΅, which is equal to π
over sin πΆ. Alternatively, that sin π΄ over π equals sin π΅ over π which equals sin πΆ over
π.

We can use either of these formulae. However, we choose to use the first one because weβre trying to find the length of a
missing side. This will minimize the amount of rearranging we need to do. We choose the second form if weβre trying to find the measure of a missing angle.

Letβs label our triangle so that it looks a little bit more like the formula
given. The side opposite the angle marked π΄ is given as lowercase π, the side opposite the
angle π΅ is lowercase π, and the side opposite angle πΆ is lowercase π.

Since weβre trying to find the length of the side π and we know the side π, weβll
use these two parts of the formula: π over sin π΄ equals π over sin πΆ. However, we are missing an angle. We do know though that the angles in a triangle add to 180 degrees. So if we subtract the given angles from 180, weβll find the measure of the angle
π΅πΆπ΄. 180 minus 54 plus 48 is equal to 78 degrees.

Once we know this, we can substitute everything that we have into our formula for the
sin rule. That gives us π over sin of 54 equals 12 over sin of 78. To solve this equation and calculate the value of π, weβll multiply both sides by
sin of 54. That gives us that π is equal to 12 over sin of 78 all multiplied by sin of 54,
which is equal to 9.925.

Correct to two decimal places, the distance between Anthony and Jennifer in feet is
9.93.

Find the distance between James and Jennifer to two decimal places.

Now, since we know the length of two of the sides in our triangle and the measure of
all three angles, at this point, we can use either the law of cosines or the law of
sines. Letβs stick with the law of sines. This time, since James and Jennifer stand at the points π΄ and πΆ, respectively,
weβre trying to calculate the length of the side that we called lowercase π.

This time, weβll use π over sin π΅ equals π over sin πΆ. Itβs preferable to use π instead of the length of the side that we called π because
weβve rounded that number. By using the length of the side π instead, weβre minimising the chance of forming
any errors from rounding too early.

Once again, letβs substitute everything we know into this formula. That gives us π over sin of 48 equals 12 over sin of 78. Once again, to solve this equation, weβll multiply both sides by sin of 48 to give us
12 over sin of 78 multiplied by sin of 48, which is equal to 9.116 and so on.

Correct to two decimal places, the distance between James and Jennifer is 9.12
feet.