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! :)
2 comments:
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.
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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