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Cosine Rule to Find the Length of an Unknown Side of a Triangle

What we’re going to do now is to look at a rule called the cosine rule. This rule is going to help us find out the unknown length of a side of a triangle, knowing the lengths of two of the sides and also knowing the size of the angle between these sides. For example, we might want to find the length of the third side of a triangle when we’ve been given the lengths of 2 of the sides as 6 and 7 units and the angle between those sides as 45 degrees. The cosine rule can also help us to find the sizes of the angles of a triangle if we’re given the lengths of all three sides. For example, if we’re told the length of the 3 sides of a triangle are 7 centimeters, 9 centimeters, and 10 centimeters, we can find the size of any of the angles we choose by using the cosine rule. It sounds useful, doesn’t it? We’ll have a look at a question like that in a future lesson.

So those are the two situations where we use the cosine rule. In this lesson, we’re going to look carefully at the first situation, where we’re given the lengths of two of the sides, and we also know the size of the angle between these sides. This is often referred to as two sides and the included angle. If we have that information, we can find the remaining side. We’re not going to discuss the maths which shows us how the cosine rule is derived. If you wish to follow this up, you could refer to a comprehensive textbook, or you’ll find it on the net.

Let’s look at the rule. It’s used if we’re given the lengths of two sides, which we’ll call a and b, and the included angle C, and we want to find the length of the opposite side c. Now, the cosine rule says c squared equals a squared plus b squared minus two times a times b times the cosine of C. As you can see, this is not an easy formula to remember, but there are some helpful hints I can give you. Can you see that it starts off exactly the same as Pythagoras’ theorem, where the square of one side equals the sum of the squares of the other two sides? So you should be able to remember the beginning. We also know that the length you’re trying to find, which is c, is the side opposite the angle we’re given the measure for, that is angle C. Notice the rule starts with the side c and finishes with the angle C, and cos also starts with a “C.” The difficult part of this formula to remember is the minus 2ab. We do know that a and b are the side measurements given, but that isn’t much of a clue. Now, that’s the part of the formula you just need to memorize.

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