James, Anthony, and Jennifer stand at three points, 𝐴, 𝐵, and 𝐶, respectively. Suppose that the measure of the angle 𝐴𝐵𝐶 is equal to 48 degrees, the measure of
the angle 𝐵𝐴𝐶 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 doesn’t need to be to scale. But it should be roughly in proportion so we can spot any mistake should they
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. We’re looking to find the length of a second side. That’s the distance between the points 𝐵 and 𝐶 on our diagram. This means we’ll need to use the law of sines.
We know that we can’t 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 𝑎 over sin 𝐴 is equal to 𝑏 over sin 𝐵, which is equal to 𝑐
over sin 𝐶. Alternatively, that sin 𝐴 over 𝑎 equals sin 𝐵 over 𝑏 which equals sin 𝐶 over
We can use either of these formulae. However, we choose to use the first one because we’re 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 we’re trying to find the measure of a missing angle.
Let’s label our triangle so that it looks a little bit more like the formula
given. The side opposite the angle marked 𝐴 is given as lowercase 𝑎, the side opposite the
angle 𝐵 is lowercase 𝑏, and the side opposite angle 𝐶 is lowercase 𝑐.
Since we’re trying to find the length of the side 𝑎 and we know the side 𝑐, we’ll
use these two parts of the formula: 𝑎 over sin 𝐴 equals 𝑐 over sin 𝐶. 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, we’ll find the measure of the angle
𝐵𝐶𝐴. 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 𝑎 over sin of 54 equals 12 over sin of 78. To solve this equation and calculate the value of 𝑎, we’ll multiply both sides by
sin of 54. That gives us that 𝑎 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
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. Let’s stick with the law of sines. This time, since James and Jennifer stand at the points 𝐴 and 𝐶, respectively,
we’re trying to calculate the length of the side that we called lowercase 𝑏.
This time, we’ll use 𝑏 over sin 𝐵 equals 𝑐 over sin 𝐶. It’s preferable to use 𝑐 instead of the length of the side that we called 𝑎 because
we’ve rounded that number. By using the length of the side 𝑐 instead, we’re minimising the chance of forming
any errors from rounding too early.
Once again, let’s substitute everything we know into this formula. That gives us 𝑏 over sin of 48 equals 12 over sin of 78. Once again, to solve this equation, we’ll 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