Video: Using the Cosine Rule to Find the Measures of Three Angles of a Triangle

Los Angeles is 1,744 miles from Chicago, Chicago is 712 miles from New York, and New York is 2,451 miles from Los Angeles. Find the angles in the triangle with its vertices as the three cities.

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Video Transcript

Los Angeles is 1,744 miles from Chicago, Chicago is 712 miles from New York, and New York is 2,451 miles from Los Angeles. Find the angles in the triangle with its vertices as the three cities.

Now, whilst a little bit of knowledge of the geography of the United States of America might be useful here, it isn’t essential to answering the problem. We can just draw a triangle using the three lengths given in the question. And if our triangle turned out to be upside down, it isn’t the end of the world. The triangle should look a little something like this, and we can add the three distances.

Now, this triangle certainly doesn’t look as if it’s a right triangle. So we’re going to need to apply either the law of sines or the law of cosines to this problem. We know all three of the side lengths, and we want to calculate each of the angles, which tells us that we should be using the law of cosines. The rearranged version of this, which is useful for calculating angles, is cos of 𝐴 equals 𝑏 squared plus 𝑐 squared minus 𝑎 squared all over two 𝑏𝑐. If you can’t remember this, you’ll have to perform the rearrangement yourself from the law of cosines in its traditional form.

In this question then, let’s use 𝐴 to represent Los Angeles, 𝐶 to represent Chicago, and 𝐵 to represent New York. We’ll use the corresponding lowercase letters to represent the opposite sides. To calculate our first angle then, that’s this angle here, we substitute the relevant values. Giving cos of 𝐴 equals 1,744 squared plus 2,451 squared minus 712 squared all over two multiplied by 1,744 multiplied by 2,451.

We can evaluate this on a calculator. And then to find the value of 𝐴, we need to use the inverse cosine function. Doing so gives 𝐴 equals 2.334 degrees. So we’ve found the first angle in the triangle. And we’ll give our answer to two decimal places.

To calculate the next angle in this triangle, which this time we’ll use angle 𝐶, we don’t need to relabel our triangle. We just need to remember that the letters 𝑏 and 𝑐 represent the two sides which enclose the angle and the letter 𝑎 represents the opposite side. So we use 1,744 and 712 for the two sides which enclose the angle and 2,451 for the side which is opposite. This gives cos of 𝐶 equals negative 0.9901. And again, applying the inverse cosine function, we find that angle 𝐶 equals 171.939 degrees.

So we found two angles in the triangle. And in fact, to find the third, we could subtract the two angles we’ve calculated from 180 degrees. But if we don’t use that method, that will be a useful check. In exactly the same way then, but this time using 712 and 2,451 as the two sides which enclose the angle and 1,744 as the opposite side, we find the measure of angle 𝐵 is 5.726 degrees.

Adding the three angles we’ve found, now each rounded to two decimal places, does indeed give 180 degrees. So we can have some confidence in our answer. The measures of the three angles in the triangle formed by these three cities, each to two decimal places then, are 2.33 degrees, 5.73 degrees, and 171.94 degrees.

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