How is the law of cosines related to the pythagorean theorem?

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How is the law of cosines related to the pythagorean theorem?

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Answer 1 · 2018-04-02T05:06:41

The Pythagorean Theorem can be stated as $a^{2} = b^{2} + c^{2}$ $a^2=b^2+c^2$ (where $a$ $a$ is the hypotenuse) and works only in right-angled triangles.

The Cosine Law is $a^{2} = b^{2} + c^{2} - 2 b c cos A$ $a^2=b^2+c^2-2bc cosA$ and works for all kinds of triangles.

Explanation:

The explanation is the really cool bit!

$A$ $A$ in the cosine rule is the angle opposite to the side $a$ $a$ .

Now, if $A = 90^{o}$ $A=90^o$ (which means the side opposite it, $a$ $a$ , is the hypotenuse), we know that ${cos 90}^{o} = 0$ $cos 90^o=0$ , so all the stuff after the minus sign in the Cosine Rule becomes $0$ $0$ ... and you will see that you just have the Pythagorean Theorem back again!

Because now you're working in a right-angled triangle.

Isn't that cool?

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How is the law of cosines related to the pythagorean theorem?

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