Law Of Cosines
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In trigonometry, the law of cosines (also known as the cosine formula or cosine rule ) relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides a , {\displaystyle a,} b , {\displaystyle b,} and c , {\displaystyle c,} opposite respective angles α , {\displaystyle \alpha ,} β , {\displaystyle \beta ,} and γ {\displaystyle \gamma } (see Fig. 1), the law of cosines states:
c 2 = a 2 + b 2 − 2 a b cos γ , a 2 = b 2 + c 2 − 2 b c cos α , b 2 = a 2 + c 2 − 2 a c cos β . {\displaystyle {\begin{aligned}c^{2}&=a^{2}+b^{2}-2ab\cos \gamma ,\\\\[3mu]a^{2}&=b^{2}+c^{2}-2bc\cos \alpha ,\\\\[3mu]b^{2}&=a^{2}+c^{2}-2ac\cos \beta .\end{aligned}}}
The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles: if γ {\displaystyle \gamma } is a right angle then cos γ = 0 , {\displaystyle \cos \gamma =0,} and the law of cosines reduces to c 2 = a 2 + b 2 . {\displaystyle c^{2}=a^{2}+b^{2}.}
The law of cosines is useful for solving a triangle when all three sides or two sides and their included angle are given.
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- Slug: law-of-cosines
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- Added: Jul 20, 2024