Friday, April 27, 2007

Trigonometry - Cosine Law, Theorem of Pythagoras

The Cosine Law works similarly to the Sine Law that I have already discussed. The Cosine Law is the general form of the Pythagorean Theorem, which itself applies strictly to right angle triangles. Therefore, this law allows us to work with any triangle. It's a bit more of an equation to remember than the Sine Law unfortunately, but it is extremely useful. Here is the equation:

c(squared) = a(squared) + b(squared) -2abCosC

(I don't see a way to type superscript text on blogger, unfortunately.)

So, as you can see, the Cosine Law is useful for finding the third side of a triangle when any 2 sides and the angle between them is known. Let's try an example:

So by the Cosine Law:
c^2 = 6^2 + 8^2 - 2(6)(8)Cos(60)
c^2 = 36 + 64 - 96Cos(60)
c^2 = 36 + 64 - 48
c^2 = 52
c = 7.2

Now that you have done that, you have obtained a complete ratio to use with the Sine Law! And now, you can find the rest of the angles to fully describe the triangle!

Sin(60)/7.2 = SinB/8
Sin(60)/7.2 = SinA/6

I'll leave those for you to solve. But that's it! Using a combination of the Cosine Law and Sine Law, you can completely solve any triangle that you are given. They are extremely powerful and useful equations!

Also, on a side note... as I mentioned at the start of this post, the Cosine Law is a generalization of the Pythagorean Theorem, which specifically applies to right angle triangles. You can see that if you are working with a right angle triangle and substitute in 90 degrees to the Cosine Law, it reduces down to the Pythagorean theorem:

c^2 = a^2 + b^2 -2abCosC
Cos(90) = 0... so the 2abCos(90) term reduces to 0
and so
c^2 = a^2 + b^2... The Pythagorean Theroem! :)


  1. Can you tell me how
    36 + 64 -96Cos(60)becomes
    36 + 64 +48

    shouldn't it be -48? or is their a bit I missed. Thanks. John.

  2. Whoops! My bad. Thanks for pointing that out, John. You're absolutely right. It should be -48. I've edited the post to show that, and the resulting side length of 7.2.


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