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

and

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

but

Cos(90) = 0... so the 2abCos(90) term reduces to 0

and so

c^2 = a^2 + b^2... The Pythagorean Theroem! :)

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

and

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

but

Cos(90) = 0... so the 2abCos(90) term reduces to 0

and so

c^2 = a^2 + b^2... The Pythagorean Theroem! :)