Sine and Cosine theorem on a triangle

Alternative formulation of the sine theorem

By transformations, the sine theorem can also be brought to the following forms:

$\frac{a}{b}=\frac{\sin \left(\alpha \right)}{\sin \left(\beta \right)}\backslash frac\{a\}\{b\}=\backslash frac\{\backslash sin\backslash left(\backslash alpha\backslash right)\}\{\backslash sin\backslash left(\backslash beta\backslash right)\}$ $\frac{a}{c}=\frac{\sin \left(\alpha \right)}{\sin \left(\gamma \right)}\backslash frac\{a\}\{c\}=\backslash frac\{\backslash sin\backslash left(\backslash alpha\backslash right)\}\{\backslash sin\backslash left(\backslash gamma\backslash right)\}$ $\frac{b}{c}=\frac{\sin \left(\beta \right)}{\sin \left(\gamma \right)}\backslash frac\{b\}\{c\}=\backslash frac\{\backslash sin\backslash left(\backslash beta\backslash right)\}\{\backslash sin\backslash left(\backslash gamma\backslash right)\}$

The Pythagorean theorem as a special case of the cosine theorem

For $\gamma ={90}^{\circ }\backslash gamma=90^\backslash circ$ we obtain a right triangle with $\cos ({90}^{\circ })=0\backslash cos(90^\backslash circ)=0$. So the Phthagorean theorem $c^2=a^2+b^{2}c^2=a^2+b^2$ is a special case of the cosine theorem.

Example

First calculate the length of side $bb$ using the sine theorem:

$\frac{a}{\sin (\alpha )}=\frac{b}{\sin \left(\beta \right)}\backslash frac\{a\}\{\backslash sin(\backslash alpha)\}=\backslash frac\{b\}\{\backslash sin\backslash left(\backslash beta\backslash right)\}$ $\Rightarrow \backslash Rightarrow$ Plug in the known values.

$\frac{\mathrm{6,1}}{\sin \left({45}^{\circ }\right)}=\frac{b}{\sin \left({55}^{\circ }\right)}\backslash frac\{6\{,\}1\}\{\backslash sin\backslash left(45^\{\backslash circ\}\backslash right)\}=\backslash frac\{b\}\{\backslash sin\backslash left(55^\{\backslash circ\}\backslash right)\}$ $\Rightarrow \backslash Rightarrow$ Solve for $bb$ .

$\Rightarrow b=\frac{6.1\cdot \sin ({55}^{\circ })}{\sin ({45}^{\circ })}=7.1\backslash Rightarrow\; b=\backslash frac\{6.1\backslash cdot\backslash sin(55^\{\backslash circ\})\}\{\backslash sin(45^\{\backslash circ\})\}=7.1$

Now calculate the length of the side $cc$ using the cosine theorem:

$c=\sqrt{a^2+b^2-2ab\cdot \cos \left(\gamma \right)}c=\backslash sqrt\{a^2+b^2-2ab\backslash cdot\backslash cos\backslash left(\backslash gamma\backslash right)\}$ $\Rightarrow \backslash Rightarrow$ Plug in the values.

$=\sqrt{{6.1}^2+{7.1}^2-2\cdot 6.1\cdot 7.1\cdot \cos \left({80}^{\circ }\right)}=8.5=\backslash sqrt\{6.1^2+7.1^2-2\backslash cdot6.1\backslash cdot7.1\backslash cdot\backslash cos\backslash left(80^\{\backslash circ\}\backslash right)\}=8.5$

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