## Related questions with answers

Question

Explain how the Pythagorean Theorem is a special case of the Law of Cosines.

Solution

VerifiedAnswered 2 years ago

Answered 2 years ago

Step 1

1 of 2To find the formula for cosine on triangles ABC and ACD we apply the Pythagorean theorem.

$\begin{align*} \triangle ADC:\\ b^{2}&=h^{2}+x^{2}\\ h^{2}&=b^{2}-x^{2}\\ \triangle ABD:\\ c^{2}&=h^{2}+(a-x)^{2}\\ c^{2}&=b^{2}-x^{2}+(a-x)^{2}\tag{$h^{2}=b^{2}-x^{2}$}\\ c^{2}&=b^{2}-x^{2}+a^{2}-2ax+x^{2}~\tag{$(a-x)^{2}=a^{2}-2ax+x^{2}$}\\ c^{2}&=b^{2}+a^{2}-2ax\\ \cos C&=\dfrac{x}{b}\\ x&=b\cdot \cos C\\ c^{2}&=b^{2}+a^{2}-2ab \cos C~\tag{for $x=b\cdot \cos C$}\\ a^{2}&=b^{2}+c^{2}-2bc \cos A\\ b^{2}&=a^{2}+c^{2}-2ac \cos B\\ \end{align*}$

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