Trigonometry/Solving Triangles Given SAS - Wikibooks

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Let's say the sides we know are b {\displaystyle b} and c {\displaystyle c} and the included angle is α {\displaystyle \alpha }

We want to find the missing side a {\displaystyle a}

We know:

b 2 + c 2 − 2 b c cos ⁡ ( α ) = a 2 {\displaystyle b^{2}+c^{2}-2bc\cos(\alpha )=a^{2}}

We have b , c {\displaystyle b,c} and α {\displaystyle \alpha } . We can therefore work out the missing side a {\displaystyle a} as

a = b 2 + c 2 − 2 b c cos ⁡ ( α ) {\displaystyle a={\sqrt {b^{2}+c^{2}-2bc\cos(\alpha )}}}

Don't be confused by the different letters we are using here to the original statement of the law of cosines! Think of the law of cosines as being a relation between two sides and the included angle, to the remaining side. We have many choices as to how we label the sides and angles.

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