8.2 Non-right triangles: law of cosines  (Page 2/8)

First, make note of what is given: two sides and the angle between them. This arrangement is classified as SAS and supplies the data needed to apply the Law of Cosines.

Each one of the three laws of cosines begins with the square of an unknown side opposite a known angle. For this example, the first side to solve for is side $b,$ as we know the measurement of the opposite angle $\beta .$

Because we are solving for a length, we use only the positive square root. Now that we know the length $b,$ we can use the Law of Sines to fill in the remaining angles of the triangle. Solving for angle $\alpha ,$ we have

The other possibility for $\alpha$ would be $\alpha =\mathrm{180°}–\mathrm{56.3°}\approx \mathrm{123.7°.}$ In the original diagram, $\alpha$ is adjacent to the longest side, so $\alpha$ is an acute angle and, therefore, $\mathrm{123.7°}$ does not make sense. Notice that if we choose to apply the Law of Cosines    , we arrive at a unique answer. We do not have to consider the other possibilities, as cosine is unique for angles between $\mathrm{0°}$ and $\mathrm{180°.}$ Proceeding with $\alpha \approx \mathrm{56.3°},$ we can then find the third angle of the triangle.

The complete set of angles and sides is

For this example, we have no angles. We can solve for any angle using the Law of Cosines. To solve for angle $\alpha ,$ we have

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