Application of the Cosine Rule (Law Of Cosines)

Alright, in my last video we covered this exact problem and we want to find the long side, c, when we are given 2 sides and an angle in between them. However, last time we did it the long way, and this time I want to show you how to solve this problem but using a short cut. And the short cut that we will use is called the "Law of Cosines" or the "Cosine Rule" which is the equation c^2 = a^2 + b^2 - 2ab*cos(theta).

So the length c, is always opposite to the angle (theta) and I will show that by drawing an arrow. And a and b represent the 2 adjacent sides to the angle. So I'll label a = 2.0m and b = 3.0m.

And before I begin plugging and chugging, I'll just mention that the cosine rule is especially useful we are given 3 sides and we want to find an angle, or if we're given 2 sides and an angle and we want to find an unknown side, which is what we have here.

So plugging our numbers in, we've got c^2 is equal to a^2, which is (2.0)^2, plus b^2 which is (3.0)^2 minus 2 times 2.0 times 3.0 times cosine of the angle 135 degrees. Now the cosine of 135 degrees is 90 past 45 degrees, or we can say that it is 45 past 90 degrees, so therefore it will have the value of negative of the 1/sqrt(2), and 3 times 2 is equal to 6, times 2 is equal to 12. So we have minus 12 times negative 1/sqrt(2) and these at the front.

So 3.0 squared is equal to 9 and 2.0 squared is equal to 4. So we have c^2 is equal to 4 + 9 - 12 * - 1/sqrt(2). 4 + 9 = 13, and then we have + 12 / sqrt(2). So this is the value for c^2.

To find c, we have to take the square-root of both sides. So if we take the square root of [13 + 12/sqrt(2)] and we plug this into a calculator, we should get roughly 4.6m.

So as you can see, the Cosine Law can really be of benefit in cutting out a lot of work when problem solving with non-right angle triangles.

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