Question Video: Solving Real-World Problems Involving the Law of Sines Mathematics

James, Anthony, and Jennifer stand at three points, 𝐴, 𝐡, and 𝐢, respectively. Suppose that π‘šβˆ π΄π΅πΆ = 48Β°, π‘šβˆ π΅π΄πΆ = 54Β°, and James is exactly 12 feet away from Anthony. Find the distance between Anthony and Jennifer, to two decimal places. Find the distance between James and Jennifer, to two decimal places.

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Video Transcript

James, Anthony, and Jennifer stand at three points, 𝐴, 𝐡, and 𝐢, respectively. Suppose that the measure of the angle 𝐴𝐡𝐢 is equal to 48 degrees, the measure of the angle 𝐡𝐴𝐢 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 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. We’re looking to find the length of a second side. That’s the distance between the points 𝐡 and 𝐢 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 π‘Ž over sin 𝐴 is equal to 𝑏 over sin 𝐡, which is equal to 𝑐 over sin 𝐢. Alternatively, that sin 𝐴 over π‘Ž equals sin 𝐡 over 𝑏 which equals sin 𝐢 over 𝑐.

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 π‘Ž, the side opposite the angle 𝐡 is lowercase 𝑏, and the side opposite angle 𝐢 is lowercase 𝑐.

Since we’re trying to find the length of the side π‘Ž and we know the side 𝑐, we’ll use these two parts of the formula: π‘Ž over sin 𝐴 equals 𝑐 over sin 𝐢. 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 𝐡𝐢𝐴. 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‘Ž over sin of 54 equals 12 over sin of 78. To solve this equation and calculate the value of π‘Ž, we’ll multiply both sides by sin of 54. That gives us that π‘Ž 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. Let’s stick with the law of sines. This time, since James and Jennifer stand at the points 𝐴 and 𝐢, respectively, we’re trying to calculate the length of the side that we called lowercase 𝑏.

This time, we’ll use 𝑏 over sin 𝐡 equals 𝑐 over sin 𝐢. It’s preferable to use 𝑐 instead of the length of the side that we called π‘Ž 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 feet.

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