Video: Using the Cosine Rule to Find an Unknown Length in a Triangle

The shown diagram represents a satellite calculating the distances and the angle. Find the distance between the two cities. Give your answer to one decimal place. Note that the diagram is not to scale.

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Video Transcript

The shown diagram represents a satellite calculating the distances and the angle. Find the distance between the two cities. Give your answer to one decimal place. Note that the diagram is not to scale.

In order to solve this problem, we need to calculate the length 𝑥, the distance between the two cities. As we have created a triangle, we can use the cosine rule to find the missing length.

The cosine rule states that 𝑎 squared is equal to 𝑏 squared plus 𝑐 squared minus two 𝑏𝑐 multiplied by cos of 𝑎. In our example, 𝑎 is 𝑥, the unknown, or the distance between the two cities, 𝑏 is 370 kilometers, 𝑐 is 350 kilometers, and the angle 𝑎 is 2.1 degrees.

Substituting these values into the formula gives us 𝑥 squared is equal to 370 squared plus 350 squared minus two multiplied by 370 multiplied by 350 multiplied by cos of 2.1. Typing the right-hand side into the calculator gives us a value of 𝑥 squared of 573.946.

Square rooting both sides of this equation gives us 𝑥 is equal to 23.957. As we’re asked to give our answer to one decimal place, 𝑥 is equal to 24.0. This means that the distance between the two cities is 24 kilometers or 24.0 kilometers.

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