Question Video: Solving Real-World Problems Involving the Law of Cosines Mathematics • 11th Grade

A plane travels 800 meters along the runway before taking off at an angle of 10°. It travels a further 1,000 meters at this angle as seen in the figure. Work out the distance of the plane from its starting point. Give your answer to 2 decimal places.

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Video Transcript

A plane travels 800 meters along the runway before taking off at an angle of 10 degrees. It travels a further 1,000 meters at this angle as seen in the figure. Work out the distance of the plane from its starting point. Give your answer to two decimal places.

Looking at the diagram, we can see that we have a triangle. We want to calculate the distance of the plane from its starting point. That’s this length here, which we can refer to as 𝑑 meters. We know the lengths of the other two sides in this triangle. They are 800 meters and 1,000 meters. And using the fact that angles on a straight line sum to 180 degrees, we can work out the size of this angle here. It’s 180 degrees minus 10 degrees, which is 170 degrees.

As this is a non-right-angled triangle, we need to answer this problem using either the law of sines or the law of cosines. So the first step is to decide which of these we need. And that will depend on the specific combination of information we’ve been given and what we want to calculate.

In this triangle, we know two sides and the included angle. And we want to calculate the third side. We recall then that this means we should be using the law of cosines. Let’s recall the law of cosines. It’s 𝑎 squared equals 𝑏 squared plus 𝑐 squared minus two 𝑏𝑐 cos 𝐴. Now, there’s no need to actually label our triangle using the letters 𝐴, 𝐵, and 𝐶. Instead, we just remember that the lowercase letters 𝑏 and 𝑐 represent the two sides we know and the capital letter 𝐴 represents the included angle.

So using 800 and 1,000 as the two side lengths 𝑏 and 𝑐 and 170 degrees as the angle 𝐴, we have the equation 𝑑 squared equals 800 squared plus 1,000 squared minus two times 800 times 1,000 times cos of 170 degrees. We can either type this directly into our calculator or it may be a good idea to break the calculation down into some stages. In either case, we arrive at 𝑑 squared equals 3,215,692.405.

Now, we must remember that this is 𝑑 squared. It isn’t 𝑑, so we aren’t finished. We have to square root in order to find the value of 𝑑. It’s a really common mistake though to forget to do this. Square rooting gives 𝑑 equals 1,793.235178. The question asks us to give our answer to two decimal places. So rounding appropriately, we’ve worked out the distance of the plane from its starting point. It’s 1,793.24 meters to two decimal places.

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